Vectors Matrices

Let's start with Vector and Matrix Algebra and build the foundation step by step. We'll cover theory, examples, and exercises. Here's an outline of how we'll proceed:

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1. Vector and Matrix Algebra

Key Topics:

• Definitions and properties of vectors and matrices
• Vector operations (addition, scalar multiplication, dot product, cross product)
• Matrix operations (addition, multiplication, transpose)
• Special matrices (identity, diagonal, symmetric, etc.)
• Applications in geometry and data representation

Learning Plan:

1. Introduction to Vectors:
2. Theory: Basics, magnitude, unit vectors

3. Practice: Solve problems involving vector addition and scalar multiplication.

4. Dot Product and Cross Product:

5. Theory: Definitions, geometric interpretations

6. Practice: Compute dot and cross products; interpret results in geometric terms.

7. Introduction to Matrices:

8. Theory: Definitions, types of matrices

9. Practice: Perform basic matrix operations like addition, scalar multiplication, and transpose.

10. Matrix Multiplication:

11. Theory: Rules and properties, row-by-column rule

12. Practice: Compute matrix products and verify properties.

13. Geometric and Practical Applications:

14. Representing transformations like rotation and scaling using matrices.

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Would you like to start with vectors? I can explain the theory, provide real-world applications, and include problems for practice.

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1. Introduction to Vectors

1.1 What is a Vector?

A vector is a mathematical object that has both magnitude (size) and direction. It is often represented as an arrow in space. For example:
- A car moving at 60 km/h in a specific direction can be represented as a vector.
- Forces acting on an object are vectors because they have magnitude and direction.

1.2 Representation of Vectors

• A vector in 2D space is written as $\mathbf{v}=[v_1,v_2]$.
• In 3D space: $\mathbf{v}=[v_1,v_2,v_3]$.

Example:
In 2D, $\mathbf{v}=[3,4]$ represents a vector with a horizontal component of 3 and a vertical component of 4.

1.3 Magnitude of a Vector

The magnitude (or length) of a vector $\mathbf{v}=[v_1,v_2,...,v_n]$ is given by:
[
|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2}
]

Example:
For $\mathbf{v}=[3,4]$,
[
|\mathbf{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5
]

1.4 Unit Vector

A unit vector has a magnitude of 1. You can find a unit vector $\hat{\mathbf{v}}$ in the direction of $\mathbf{v}$ by dividing it by its magnitude:
[
\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}
]

Example:
For $\mathbf{v}=[3,4]$,
[
\hat{\mathbf{v}} = \frac{[3, 4]}{5} = [\frac{3}{5}, \frac{4}{5}]
]

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1.5 Operations on Vectors

Addition of Vectors

If $\mathbf{u}=[u_1,u_2]$ and $\mathbf{v}=[v_1,v_2]$, then:
[
\mathbf{u} + \mathbf{v} = [u_1 + v_1, u_2 + v_2]
]

Example:
$\mathbf{u}=[2,3]$, $\mathbf{v}=[4,1]$:
[
\mathbf{u} + \mathbf{v} = [2 + 4, 3 + 1] = [6, 4]
]

Scalar Multiplication

If $c$ is a scalar and $\mathbf{v}=[v_1,v_2]$, then:
[
c \mathbf{v} = [c v_1, c v_2]
]

Example:
$c=3$, $\mathbf{v}=[2,3]$:
[
3 \mathbf{v} = [3 \cdot 2, 3 \cdot 3] = [6, 9]
]

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Practice Problems

1. Find the magnitude of the vector $\mathbf{v}=[5,12]$.
2. Normalize $\mathbf{v}=[6,8]$ to find its unit vector.
3. Add the vectors $\mathbf{u}=[1,2]$ and $\mathbf{v}=[3,4]$.
4. Multiply the vector $\mathbf{v}=[7,9]$ by the scalar $c=0.5$.

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Would you like solutions for these, or should we explore more advanced topics like the dot product?

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Solutions to Practice Problems

1. Find the magnitude of the vector $\mathbf{v}=[5,12]$:

Using the formula for magnitude:
[
|\mathbf{v}| = \sqrt{v_1^2 + v_2^2}
]
Substitute $v_1=5$ and $v_2=12$:
[
|\mathbf{v}| = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
]

Answer: The magnitude is 13.

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2. Normalize $\mathbf{v}=[6,8]$ to find its unit vector:

First, find the magnitude of $\mathbf{v}$:
[
|\mathbf{v}| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
]

The unit vector is:
[
\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|} = \frac{[6, 8]}{10} = [\frac{6}{10}, \frac{8}{10}] = [0.6, 0.8]
]

Answer: The unit vector is $[0.6,0.8]$.

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3. Add the vectors $\mathbf{u}=[1,2]$ and $\mathbf{v}=[3,4]$:

$$\mathbf{u}+\mathbf{v}=[1+3,2+4]=[4,6]$$

Answer: $\mathbf{u}+\mathbf{v}=[4,6]$.

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4. Multiply the vector $\mathbf{v}=[7,9]$ by the scalar $c=0.5$:

$$c\mathbf{v}=[0.5\cdot 7,0.5\cdot 9]=[3.5,4.5]$$

Answer: $0.5\mathbf{v}=[3.5,4.5]$.

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Next Steps: Dot Product and Geometric Applications

Would you like to proceed with the dot product and its applications, or explore real-world use cases of these operations?

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2. Dot Product and Cross Product of Vectors

2.1 Dot Product

The dot product of two vectors $\mathbf{u}=[u_1,u_2,...,u_n]$ and $\mathbf{v}=[v_1,v_2,...,v_n]$ is defined as:
[
\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + ... + u_nv_n = \sum_{i=1}^n u_i v_i
]

Geometric Interpretation

The dot product can also be written as:
[
\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos\theta
]
where $\theta$ is the angle between the vectors.
- If $\theta =0^{\circ }$: $\cos \theta =1$, and the vectors are parallel.
- If $\theta ={90}^{\circ }$: $\cos \theta =0$, and the vectors are orthogonal (perpendicular).

Properties of the Dot Product

1. Commutative: $\mathbf{u}\cdot \mathbf{v}=\mathbf{v}\cdot \mathbf{u}$
2. Distributive: $\mathbf{u}\cdot (\mathbf{v}+\mathbf{w})=\mathbf{u}\cdot \mathbf{v}+\mathbf{u}\cdot \mathbf{w}$
3. Scalar multiplication: $(c\mathbf{u})\cdot \mathbf{v}=c(\mathbf{u}\cdot \mathbf{v})$

Example Calculation

If $\mathbf{u}=[2,3]$ and $\mathbf{v}=[4,1]$:
[
\mathbf{u} \cdot \mathbf{v} = (2 \cdot 4) + (3 \cdot 1) = 8 + 3 = 11
]

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2.2 Cross Product (Only for 3D Vectors)

The cross product of two 3D vectors $\mathbf{u}=[u_1,u_2,u_3]$ and $\mathbf{v}=[v_1,v_2,v_3]$ is a vector defined as:
[
\mathbf{u} \times \mathbf{v} =
$$\left|\begin{array}{ccccccc}\mathbf{i}& \mathbf{j}& \mathbf{k}{u}_1& u_2& u_3{v}_1& v_2& v_3\end{array}\right|$$
= \mathbf{i}(u_2v_3 - u_3v_2) - \mathbf{j}(u_1v_3 - u_3v_1) + \mathbf{k}(u_1v_2 - u_2v_1)
]
where $\mathbf{i},\mathbf{j},\mathbf{k}$ are the unit vectors along the $x$, $y$, and $z$-axes.

Geometric Interpretation

The cross product results in a vector that is:
1. Perpendicular to both $\mathbf{u}$ and $\mathbf{v}$.
2. Has a magnitude equal to the area of the parallelogram formed by $\mathbf{u}$ and $\mathbf{v}$:
[
|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}| |\mathbf{v}| \sin\theta
]

Example Calculation

If $\mathbf{u}=[1,2,3]$ and $\mathbf{v}=[4,5,6]$:
[
\mathbf{u} \times \mathbf{v} =
$$\left|\begin{array}{ccccccc}\mathbf{i}& \mathbf{j}& \mathbf{k}1& 2& 34& 5& 6\end{array}\right|$$
= \mathbf{i}((2)(6) - (3)(5)) - \mathbf{j}((1)(6) - (3)(4)) + \mathbf{k}((1)(5) - (2)(4))
]
[
= \mathbf{i}(12 - 15) - \mathbf{j}(6 - 12) + \mathbf{k}(5 - 8)
]
[
= -3\mathbf{i} + 6\mathbf{j} - 3\mathbf{k}
]
[
\mathbf{u} \times \mathbf{v} = [-3, 6, -3]
]

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Practice Problems

1. Find the dot product of $\mathbf{u}=[3,4]$ and $\mathbf{v}=[5,2]$.
2. If $\mathbf{u}=[1,0,-1]$ and $\mathbf{v}=[2,3,1]$, compute $\mathbf{u}\cdot \mathbf{v}$ and interpret it.
3. Calculate the cross product of $\mathbf{u}=[2,3,4]$ and $\mathbf{v}=[5,6,7]$.
4. Given $\mathbf{u}=[1,1]$ and $\mathbf{v}=[1,-1]$, determine if the vectors are orthogonal.

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Let me know if you'd like solutions to these or if you'd prefer to move to matrix operations next!

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Solutions to Practice Problems

1. Add the matrices:

$$A=\left[\begin{array}{cc}2& 4\\ 6& 8\end{array}\right],B=\left[\begin{array}{cc}1& 3\\ 5& 7\end{array}\right]$$

Matrix addition is performed element-wise:
[
A + B = $$\left[\begin{array}{ccc}2+1& 4+36+5& 8+7\end{array}\right]$$ =
$$\left[\begin{array}{ccc}3& 711& 15\end{array}\right]$$
]

Answer:
[
A + B = $$\left[\begin{array}{ccc}3& 711& 15\end{array}\right]$$
]

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2. Multiply the matrix by a scalar:

$$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],c=-1$$

Scalar multiplication is performed element-wise:
[
cA = $$\left[\begin{array}{ccc}-1\cdot 1& -1\cdot 2-1\cdot 3& -1\cdot 4\end{array}\right]$$ =
$$\left[\begin{array}{ccc}-1& -2-3& -4\end{array}\right]$$
]

Answer:
[
cA = $$\left[\begin{array}{ccc}-1& -2-3& -4\end{array}\right]$$
]

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3. Compute the product of the matrices:

$$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$$

Matrix multiplication is computed as:
[
C = AB = $$\left[\begin{array}{ccc}(1\cdot 5+2\cdot 7)& (1\cdot 6+2\cdot 8)(3\cdot 5+4\cdot 7)& (3\cdot 6+4\cdot 8)\end{array}\right]$$
]

Calculate element-wise:
[
C = $$\left[\begin{array}{ccc}5+14& 6+1615+28& 18+32\end{array}\right]$$ =
$$\left[\begin{array}{ccc}19& 2243& 50\end{array}\right]$$
]

Answer:
[
C = $$\left[\begin{array}{ccc}19& 2243& 50\end{array}\right]$$
]

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4. Find the transpose of the matrix:

$$A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$$

The transpose of $A$ is obtained by swapping rows and columns:
[
A^T = $$\left[\begin{array}{cccc}1& 42& 53& 6\end{array}\right]$$
]

Answer:
[
A^T = $$\left[\begin{array}{cccc}1& 42& 53& 6\end{array}\right]$$
]

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Next Steps: Determinants and Matrix Inverses

Would you like to proceed with the determinants and how they relate to matrix inverses, or explore real-world applications of these concepts?

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4. Determinants and Matrix Inverses

4.1 Determinant of a Matrix

The determinant is a scalar value that is computed from a square matrix. It provides essential information about the matrix, such as whether it is invertible and its geometric properties.

4.1.1 Determinant of a 2x2 Matrix

For a $2\times 2$ matrix:
[
A = $$\left[\begin{array}{ccc}a& bc& d\end{array}\right]$$
]
The determinant is:
[
\text{det}(A) = |A| = ad - bc
]

Example:
For $A=\left[\begin{array}{cc}3& 8\\ 4& 6\end{array}\right]$:
[
\text{det}(A) = (3)(6) - (8)(4) = 18 - 32 = -14
]

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4.1.2 Determinant of a 3x3 Matrix

For a $3\times 3$ matrix:
[
A = $$\left[\begin{array}{ccccccc}a& b& cd& e& fg& h& i\end{array}\right]$$
]
The determinant is computed as:
[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
]

Example:
For $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$:
[
\text{det}(A) = 1((5)(9) - (6)(8)) - 2((4)(9) - (6)(7)) + 3((4)(8) - (5)(7))
]
[
\text{det}(A) = 1(45 - 48) - 2(36 - 42) + 3(32 - 35)
]
[
\text{det}(A) = -3 + 12 - 9 = 0
]

This matrix is not invertible because its determinant is 0.

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4.2 Inverse of a Matrix

The inverse of a square matrix $A$, denoted $A^{-1}$, is a matrix such that:
[
A A^{-1} = A^{-1} A = I
]
where $I$ is the identity matrix.

4.2.1 Inverse of a 2x2 Matrix

For a $2\times 2$ matrix:
[
A = $$\left[\begin{array}{ccc}a& bc& d\end{array}\right]$$
]
The inverse is given by:
[
A^{-1} = \frac{1}{\text{det}(A)} $$\left[\begin{array}{ccc}d& -b-c& a\end{array}\right]$$, \quad \text{if } \text{det}(A) \neq 0
]

Example:
For $A=\left[\begin{array}{cc}3& 8\\ 4& 6\end{array}\right]$, $\text{det}(A)=-14$:
[
A^{-1} = \frac{1}{-14} $$\left[\begin{array}{ccc}6& -8-4& 3\end{array}\right]$$ = $$\left[\begin{array}{ccc}-\frac{3}{7}& \frac{4}{7}\frac{2}{7}& -\frac{3}{14}\end{array}\right]$$
]

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4.2.2 Conditions for Invertibility

A square matrix $A$ is invertible if and only if $\text{det}(A)\ne 0$.

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Practice Problems

1. Find the determinant of the matrix:
   [
   A = $$\left[\begin{array}{ccc}2& 31& 4\end{array}\right]$$
   ]

2. Compute the determinant of:
   [
   B = $$\left[\begin{array}{ccccccc}1& 0& 23& 4& 56& 7& 8\end{array}\right]$$
   ]

3. Check if the matrix is invertible and find its inverse if possible:
   [
   C = $$\left[\begin{array}{ccc}1& 23& 4\end{array}\right]$$
   ]

4. Prove that a matrix with determinant $0$ is singular (not invertible) by finding the determinant of:
   [
   D = $$\left[\begin{array}{ccc}1& 22& 4\end{array}\right]$$
   ]

Would you like solutions to these problems, or should we move to systems of linear equations and their solutions?

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