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1.3 Vector Equations

Vectors can be represented graphically in a coordinate system or algebraically as column matrices.

A 2D coordinate system showing vectors u at (1,2) and v at (1,-1) originating from the origin.
\[ \vec{u} = 1\vec{i} + 2\vec{j} = \langle 1, 2 \rangle \]\[ = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \]

matrix w/ just one column → column vector

\[ \vec{v} = \begin{bmatrix} +1 \\ -1 \end{bmatrix} \]

Vector Addition and Subtraction

\[ \vec{u} + \vec{v} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} + \begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix} \]

The sum \( \vec{u} + \vec{v} \) can be visualized using the parallelogram rule.

Parallelogram rule for vector addition showing u plus v as the diagonal.
\[ \vec{u} - \vec{v} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} - \begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 0 \\ 3 \end{bmatrix} \]

The difference \( \vec{u} - \vec{v} \) is the vector from the tip of \( \vec{v} \) to the tip of \( \vec{u} \).

Vector subtraction diagram showing u minus v.
\[ 3\vec{u} + 2\vec{v} \]\[ = 3\begin{bmatrix} 1 \\ 2 \end{bmatrix} + 2\begin{bmatrix} 1 \\ -1 \end{bmatrix} \]
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Any vector in the form \( a\vec{u} + b\vec{v} \) is called a linear combination of \( \vec{u} \) and \( \vec{v} \).

\[ \vec{u} = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \quad \vec{v} = \begin{bmatrix} 3 \\ 5 \\ 2 \end{bmatrix} \quad \vec{w} = \begin{bmatrix} -1 \\ 4 \\ -3 \end{bmatrix} \]

Linear combination of these:

\[ x_1 \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} + x_2 \begin{bmatrix} 3 \\ 5 \\ 2 \end{bmatrix} + x_3 \begin{bmatrix} -1 \\ 4 \\ -3 \end{bmatrix} \]\[ = \begin{bmatrix} x_1 \\ 0 \\ x_1 \end{bmatrix} + \begin{bmatrix} 3x_2 \\ 5x_2 \\ 2x_2 \end{bmatrix} + \begin{bmatrix} -x_3 \\ 4x_3 \\ -3x_3 \end{bmatrix} \]\[ = \begin{bmatrix} x_1 + 3x_2 - x_3 \\ 5x_2 + 4x_3 \\ x_1 + 2x_2 - 3x_3 \end{bmatrix} \text{ is a vector that is linear combo of } \vec{u}, \vec{v}, \vec{w} \]

looks like a system

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Example: Systems of Linear Equations

Consider the following system of linear equations:

\[\begin{aligned} 4x_1 + x_2 + 3x_3 &= 9 \\ x_1 - 7x_2 - 2x_3 &= 2 \\ 8x_1 + 6x_2 - 5x_3 &= 15 \end{aligned}\]

We can rewrite this system as a single vector equation:

\[\begin{bmatrix} 4x_1 + x_2 + 3x_3 \\ x_1 - 7x_2 - 2x_3 \\ 8x_1 + 6x_2 - 5x_3 \end{bmatrix} = \begin{bmatrix} 9 \\ 2 \\ 15 \end{bmatrix}\]

This is equivalent to the linear combination of column vectors:

\[x_1 \begin{bmatrix} 4 \\ 1 \\ 8 \end{bmatrix} + x_2 \begin{bmatrix} 1 \\ -7 \\ 6 \end{bmatrix} + x_3 \begin{bmatrix} 3 \\ -2 \\ -5 \end{bmatrix} = \begin{bmatrix} 9 \\ 2 \\ 15 \end{bmatrix}\]

This is equivalent to the augmented matrix:

\[\left[ \begin{array}{ccc|c} 4 & 1 & 3 & 9 \\ 1 & -7 & -2 & 2 \\ 8 & 6 & -5 & 15 \end{array} \right]\]

So, solving a system is the same as asking if the right hand side vector is a linear combo of the columns of the coefficient matrix.

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Example: Linear Combinations

Is \(\begin{bmatrix} 2 \\ -1 \\ 6 \end{bmatrix}\) a linear combo of \(\begin{bmatrix} 1 \\ -2 \\ 0 \end{bmatrix}\), \(\begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}\), and \(\begin{bmatrix} 5 \\ -6 \\ 8 \end{bmatrix}\)?

Can we find \(x_1, x_2, x_3\) such that:

\[x_1 \begin{bmatrix} 1 \\ -2 \\ 0 \end{bmatrix} + x_2 \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix} + x_3 \begin{bmatrix} 5 \\ -6 \\ 8 \end{bmatrix} = \begin{bmatrix} 2 \\ -1 \\ 6 \end{bmatrix}\]

This is the same as the system of equations:

\[\begin{aligned} x_1 + 0x_2 + 5x_3 &= 2 \\ -2x_1 + x_2 - 6x_3 &= -1 \\ 0x_1 + 2x_2 + 8x_3 &= 6 \end{aligned}\]

And the corresponding augmented matrix:

\[\begin{bmatrix} 1 & 0 & 5 & 2 \\ -2 & 1 & -6 & -1 \\ 0 & 2 & 8 & 6 \end{bmatrix}\]
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\[ \sim \dots \sim \begin{bmatrix} 1 & 0 & 5 & 2 \\ 0 & 1 & 4 & 3 \\ 0 & 0 & 0 & 0 \end{bmatrix} \]

\( x_3 \) is free

\[ x_2 + 4x_3 = 3 \rightarrow x_2 = 3 - 4x_3 \]\[ x_1 + 5x_3 = 2 \rightarrow x_1 = 2 - 5x_3 \]

At least one solution exists, so

\[ \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix} \text{ is a linear combo of } \begin{bmatrix} 1 \\ -2 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 5 \\ -6 \\ 8 \end{bmatrix} \]

One way: \( x_3 = 1 \) (arbitrary)

\( x_2 = -1 \)

\( x_1 = -3 \)

So

\[ -3 \begin{bmatrix} 1 \\ -2 \\ 0 \end{bmatrix} - \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix} + \begin{bmatrix} 5 \\ -6 \\ 8 \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix} \]
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Example

Is \( \begin{bmatrix} 8 \\ 1 \\ 1 \end{bmatrix} \) a linear combo of \( \begin{bmatrix} 0 \\ 2 \\ 4 \end{bmatrix}, \begin{bmatrix} +1 \\ -3 \\ -8 \end{bmatrix}, \begin{bmatrix} -4 \\ 2 \\ 12 \end{bmatrix} \)?

\[ \begin{bmatrix} 0 & 1 & -4 & 8 \\ 2 & -3 & 2 & 1 \\ 4 & -8 & 12 & 1 \end{bmatrix} \]
\[ \begin{bmatrix} 2 & -3 & 2 & 1 \\ 0 & 1 & -4 & 8 \\ 4 & -8 & 12 & 1 \end{bmatrix} \]
\[ \begin{bmatrix} 2 & -3 & 2 & 1 \\ 0 & 1 & -4 & 8 \\ 0 & -2 & 8 & -1 \end{bmatrix} \]
\[ \begin{bmatrix} 2 & -3 & 2 & 1 \\ 0 & 1 & -4 & 8 \\ 0 & 0 & 0 & 15 \end{bmatrix} \]

no solution

no, not a linear combo.

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Linear Combinations and Span

Let

\[ \vec{a}_1 = \begin{bmatrix} 1 \\ -2 \\ 0 \end{bmatrix}, \quad \vec{a}_2 = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}, \quad \vec{a}_3 = \begin{bmatrix} 5 \\ -6 \\ 8 \end{bmatrix} \]

And from earlier, we know \( \begin{bmatrix} 2 \\ -1 \\ 6 \end{bmatrix} \) is a linear combo of \( \vec{a}_1, \vec{a}_2, \vec{a}_3 \).

Definition of Span

The set of all possible linear combos of \( \vec{a}_1, \vec{a}_2, \vec{a}_3 \) is called the subset spanned by \( \vec{a}_1, \vec{a}_2, \vec{a}_3 \), written as \( \text{span} \{ \vec{a}_1, \vec{a}_2, \vec{a}_3 \} \).

So, \( \begin{bmatrix} 2 \\ -1 \\ 6 \end{bmatrix} \) is in the \( \text{span} \{ \vec{a}_1, \vec{a}_2, \vec{a}_3 \} \).

Examples

  • What is \( \text{span} \left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\} \)? xy plane
  • What is \( \text{span} \left\{ \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\} \)? y-axis
  • What is \( \text{span} \left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \end{bmatrix} \right\} \)? x-axis, same as \( \text{span} \left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right\} \)
A 2D Cartesian coordinate system with x and y axes. Vector (0,1) points up the y-axis and (1,0) points right on the x-axis.