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For the matrices in Exercises $17-20,(a)$ find $k$ such that Nul $A$ is a subspace of $\mathbb{R}^{k},$ and $(b)$ find $k$ such that $\operatorname{Col} A$ is a subspace of $\mathbb{R}^{k} .$ $$A=\left[\begin{array}{rr}{2} & {-6} \\ {-1} & {3} \\ {-4} & {12} \\ {3} & {-9}\end{array}\right]$$

a. 2 b. 4

Calculus 3

Chapter 4

Vector Spaces

Section 2

Null Spaces, Column Spaces, and Linear Transformations

Vectors

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in this video, we're starting out with a four by two matrix A that's indicated here. Now when you know for this particular matrix A. We conform to interesting sub spaces. The first up space is the null space of a Let's write out what the null space of a is specifically. It's the set of all vectors X such that the matrix equation eight times X equals zero is satisfied. Or you could say it's all vectors X that are mapped to zero by the transformation X goes to a X, so we want to determine next the following. We know that the null space of a is a subspace of the following. So no, a is a subspace of our something. So we want to determine what should go here. Will it be? Are for or will it be are too? One way to determine what their appropriate value is is to look very closely up This matrix multiplication, the Matrix A is of size for by two and the zero vector is going to be of size four by one. That tells us something about the vector X. What's right down the dimensions here. Well, first it has to be a to, in order for this value to match up and allow for the multiplication to be defined, and it has to be of size one. So we match this outer dimension here. So the Vector X, which is what the null space consists of, has two entries, and that tells us no lays a subspace of our two. So that's one very interesting and important subspace that comes from this matrix. Let's change gears and talk about the column space of a Indicated this way. Well, the column space of a we Know by words is equal to the span of the columns of a and the span of the columns of a would be linear combinations of this column and this column. Such linear combinations give us new vectors, which still contain four entries, but that employees next that the calm space of A is a subspace of altogether are four since we're spanning vectors with four entries, so to pause for a moment before we end this example when we're considering the null space and we want to know that dimension, or rather which subspace it belongs to, we look to the number of columns. So are two for this matrix. If you're concerned about the column space, we look to the number of rows and that will tell us where the calm space is contained in.

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