## Example 1

Let $$T:\mathbb{R}^2\rightarrow \mathbb{R}^3$$ be a linear transformation such that $$T\left(\begin{bmatrix} 2 \\ 1 \end{bmatrix}\right) = \begin{bmatrix} 1 \\ 0 \\ 1\end{bmatrix}$$ and $$T\left(\begin{bmatrix} 1 \\ 1\end{bmatrix}\right) = \begin{bmatrix} 0 \\ -1 \\ 1\end{bmatrix}$$. What is $$T\left(\begin{bmatrix} 0 \\ -1\end{bmatrix}\right)$$?

## Example 2

Consider the linear transformation $$T:\mathbb{R}^4 \rightarrow \mathbb{R}^{2\times 2}$$ given by $$T \left( \left[{\begin{array}{c} a \\ b \\ c \\ d\\ \end{array} } \right] \right) = \left[ \begin{array}{cc} 2a+b+c & a-d \\ a+b+c-d & 0 \\ \end{array} \right].$$

1. Determine a basis for the range of $$T$$.

2. Give a basis for the kernel of $$T$$.

3. Is $$T$$ surjective? Injective?

## Example 3

Let $$\mathbb{F}$$ denote the field $$GF(2)$$. Let $$T:\mathbb{F}^4 \rightarrow \mathbb{F}^3$$ be a linear transformation given by $$T\left(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}\right) = \begin{bmatrix} x_1 + x_2 + x_4 \\ x_1 + x_3 + x_4 \\ x_2 + x_3 + x_4 \end{bmatrix}$$. Give a basis for the kernel of $$T$$.

## Example 4

Let $$T:\mathbb{R}^3\rightarrow \mathbb{R}^3$$ be a linear transformation given by $$T\left(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\right) = \begin{bmatrix} x_1 + x_2 + x_3 \\ -x_1 -x_2 + x_3 \\ x_2 - x_3\end{bmatrix}$$. Determine if $$T$$ is invertible.

## Example 5

Let $$T:\mathbb{R}^3\rightarrow \mathbb{R}^2$$ be a linear transformation given by $$T\left(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\right) = \begin{bmatrix} x_1 + x_2 \\ -x_1 -x_2 + x_3\end{bmatrix}$$. Let $$\Gamma = \left (\begin{bmatrix} 1 \\ 0 \\ 0\end{bmatrix}, \begin{bmatrix} 1 \\ -1 \\ 0\end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1\end{bmatrix} \right )$$ be an ordered basis for $$\mathbb{R}^3$$ and let $$\Omega = \left ( \begin{bmatrix} 1 \\ 1\end{bmatrix}, \begin{bmatrix} -1 \\ 1\end{bmatrix} \right )$$ be an ordered basis for $$\mathbb{R}^2$$. Find $$[T]_\Gamma^\Omega$$.

## Example 6

Let $$P_2$$ denote the vector space of polynomials in $$x$$ with real coefficients having degree at most $$2$$. Let $$T:P_2\rightarrow \mathbb{R}^3$$ be a linear transformation given by $$T (ax^2 + bx + c) = \begin{bmatrix} 2a-b \\ b+2c \\ 0\end{bmatrix}$$. Determine the dimension of the kernel of $$T$$.