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Determine if $\mathbf{w}=\left[\begin{array}{r}{1} \\ {3} \\ {-4}\end{array}\right]$ is in Nul $A,$ where $$A=\left[\begin{array}{rrr}{3} & {-5} & {-3} \\ {6} & {-2} & {0} \\ {-8} & {4} & {1}\end{array}\right]$$

$w$ is in the null space of $A$

Calculus 3

Chapter 4

Vector Spaces

Section 2

Null Spaces, Column Spaces, and Linear Transformations

Vectors

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in this example, we have a matrix that's provided and a vector us well. And Arkle here is to determine if the vector W is in the null space of this matrix. A. Let's start out by recalling with the null space it is, we right know a to refer to the null space of a and it is a set. It's a set of all vectors X such that eight times x results in the zero vector. So to check if W is in the null space of a really, all we're concerned about is whether or not this equation is true by definition. So let's jump to focusing on this equation. Are X is gonna play the role of W. In this case, and so we're taking a times w Let me copy a first. It's 36 negative. Eight negative five negative to four and negative 301 We're multiplying by the Vector W, which is 13 and negative for, and we're going to check if we get zero vector. First we take the first entry, which is the column here, multiplied by the corresponding trees and this row, producing three minus 15 plus 12 and this is zero altogether. So if we get two more zeros were are in the null space. So we'll take the same column multiplied by this row. We have six then minus six plus zero. So this is getting exciting. We almost have for certainty that this is in the null space, but it all comes down to this last entry. So multiply this row by the column vector and will get negative eight plus 12 minus four, which is zero as well. So we have a zero. Here, let me fix that. Okay, that looks better. This is the zero factor. So we just chuck shown a times w equals a zero vector just as in this equation here. But don't forget, we have to state our conclusions in linear algebra. So our conclusion is that w is in no a for this particular matrix a

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