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[M] Determine if $\mathbf{y}$ is in the subspace of $\mathbb{R}^{4}$ spanned by the columns of $A,$ where$\mathbf{y}=\left[\begin{array}{r}{-4} \\ {-8} \\ {6} \\ {-5}\end{array}\right], \quad A=\left[\begin{array}{rrr}{3} & {-5} & {-9} \\ {8} & {7} & {-6} \\ {-5} & {-8} & {3} \\ {2} & {-2} & {-9}\end{array}\right]$

Therefore, the vector $\mathbf{y}$ is a linear combination of given three $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}$Thus, the vector $\mathbf{y}$ is in the subspace of $\mathbb{R}^{4}$ spanned by the vectors $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}$

Calculus 3

Chapter 4

Vector Spaces

Section 1

Vector Spaces and Subspaces

Vectors

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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in this video only solving problem. Number 36 of section 4.1 Grand word. Uh, we're asked to determine if why is in the subspace of our four spend by the columns of a where why is, um, why is already in our for a zit has four entries given here, and but we need to see if it's spend by these specific columns in our four. So in a better part of the matrix, a first we need to set up in augmented matrix here, um, to solve this system. So we just combine the two matrices are to, uh, the Matrix and the vector. So here we have 38 negative five to as the first column. Negative five seven. Negative eight. And negative, too, was the second. Call them negative. Nine Negative 63 Negative nine. Is that their call? And then a line in between. Negative four. Negative eight six and negative five. As the last column as this is the B column in the Matrix equation, X equals B in resolving for a set of scale er's X, where any where x one multiplied by the first column and x two multiplied by the second call in the next three multiplied. But third call them would equal why so this season are you have calculator and to make your life easier sense Doing this by hand would take a long time. But, um, so you get if you saw this, you should get and put it in our area. We're reduced to ash lawn for me. Should get negative one on DDE 3/5 on zero. So, um, we solved for three values your negative 1 52 53 5th where? Negative 1/5 times. What's called this? The one or a one 81 83. So the night of 1/5 time's a one plus negative to fifth times a two and was 3 50 times a three equals. Why? So this, um since there is a solution to this system, you know that we know that, um, Why is spanned by these columns the columns of this matrix, and are for

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