Linear Algebra Examples

$[\begin{array}{c} d + j^{t} & g + k^{t} & h + l^{t} \\ d + j & g + k & h + l \\ n & o & p \end{array}] = (1 - t^{2}) [\begin{array}{c} d & g & h \\ j & k & l \\ n & o & p \end{array}]$

Step 1

The kernel of a transformation is a vector that makes the transformation equal to the zero vector (the pre-image of the transformation).

$[\begin{array}{c} 1 - t^{2} \end{array}] = 0$

Step 2

Create a system of equations from the vector equation.

$1 - t^{2} = 0$

Step 3

Subtract $1$ from both sides of the equation.

$- t^{2} = - 1$

Step 4

Write the system of equations in matrix form.

$[\begin{array}{c} - 1 & - 1 \end{array}]$

Step 5

Find the reduced row echelon form of the matrix.

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Multiply each element of $R_{1}$ by $- 1$ to make the entry at $1, 1$ a $1$ .

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Multiply each element of $R_{1}$ by $- 1$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} - - 1 & - - 1 \end{array}]$

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 1 \end{array}]$

Step 6

Use the result matrix to declare the final solutions to the system of equations.

$x = 1$

Step 7

This expression is the solution set for the system of equations.

${(1)}$

Step 8

Decompose a solution vector by re-arranging each equation represented in the row-reduced form of the augmented matrix by solving for the dependent variable in each row yields the vector equality.

$X = [\begin{array}{c} x \end{array}] = [\begin{array}{c} 1 \end{array}]$

Step 9

The null space of the set is the set of vectors created from the free variables of the system.

${[\begin{array}{c} 1 \end{array}]}$

Step 10

The kernel of $M$ is the subspace ${[\begin{array}{c} 1 \end{array}]}$ .

$K (M) = {[\begin{array}{c} 1 \end{array}]}$