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I'm having trouble understanding what it means to find the kernel of a matrix.

I have a matrix as follows and I need to determine the kernel.

$\begin{bmatrix}2 & 4 & 5 \\ 1 & 2 & 3 \\ -1 & -1 & -3 \end{bmatrix}$

I'm able to get the augmented matrix down to echelon form as follows.

$\begin{bmatrix}1 & 2 & 3 & 0 \\ 0 & 5 & 4 & 0\end{bmatrix}$

Where do I take it from here? What would the form of the solution be?

Would I just write a couple of equations like the following two?

$x_1+2x_2+3x_3=0$

$0x_1+5x_2+4x_3=0$

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    Yes. You see the solutions will depend on, say $x_3$, and setting $x_3=t$, you will be able to write a solution as $t\times$ some fixed vector. – 2017-02-27

2 Answers 2

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I found that this Khan Academy video was the best resource for helping me understand this problem more.

In the end I wrote the following for my answer.

$x_1=-\frac{7}{5}x_3+\frac{1}{5}x_4$

$x_2=-\frac{4}{5}x_3-\frac{3}{5}x_4$

I wasn't certain how to express this with $t$, as Dovah-king and the video mentioned, because we have two free variables, unlike the example in the video.

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Finding the kernel of a matrix $A$ is finding the set of vectors that, when multiplied by A, result in the vector $0$. (It is easy to verify that this set of vectors is a vector space) Mathematically speaking, you must solve the equation: $Ax = 0$, where $x$ is an vector. Note that this equation might have one solution or infinite solutions. In the second case, you'll find, while solving the system, a free variable, for example, $x_2$. Then, your other vectors can depend of this variable. You will find a free variable if you find something like $0 x_2 = 0$. Notice that you can have more than one free variable. The number of free variables is also the dimension of the null space of a transformation $T$ defined by $T(x) = Ax$.

Notice that, in your case, you have that your 3rd line is null. This means that $x_3$ is your free variable. Call it $t$ and write the solution in terms of it.