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Could a set of three vectors in $\mathbb{R}^{4}$ span all of $\mathbb{R}^{4} ?$ Explain. What about $n$ vectors in $\mathbb{R}^{m}$ when $n$ is less than $m ?$

Same thing for $n$ vectors in $\mathbb{R}^{m}$ . Than matrix can have at most $n$ pivot$\left.\text { columns, so it can span } \mathbb{R}^{n} \text { but not } \mathbb{R}^{m} \text { (because } n<m\right)$ .

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 4

The Matrix Equation Ax D b

Introduction to Matrices

Campbell University

Oregon State University

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

02:39

Let $S$ be the subspace of…

03:58

Recall that three vectors …

03:03

01:09

Give a geometric descripti…

02:00

Find the dimension of the …

02:57

03:12

02:36

01:34

Find three mutually orthog…

05:27

Determine whether the give…

Okay, so for a problem 32. Um um, we're asking coulda set off three vectors ing our force been all our four. So the answer. Excuse me? The answer is definitely no or, um yeah, that the answer is definitely no. The reason is that if we want, we want to want want some some base factors to spend. They are for the least number of factors we need is four vectors. So you can just think about, uh, such a vector like, um 1000 100 And I guess that I think about such a such a basis, like 0100 and 0010 and 0001 So these four vectors spend our four. But if we delete any one of them, then the best that we can get is the are three women are three space because we lose a will, use a vector that and will define a one more. One more dimension. So to find out the span off of her four dimensional space, the least vectors we need is four vectors. So three directors will not be enough. So what about, uh what about in vectors for all right, our m within smaller than m again This again Not It's not possible. The reason is the same. We need at least I mean at least am vectors to spend the whole space off our r m. Because with the dimension of em, we we have to we have to make m vectors to divine am dimensions. So if we only have in vectors, then we cannot divide every single dimension off our M. So this is also not possible.

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