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Exercises $17-20$ refer to the matrices $A$ and $B$ below. Make appropriate calculations that justify your answers and mention an appropriate theorem.$$A=\left[\begin{array}{rrrr}{1} & {3} & {0} & {3} \\ {-1} & {-1} & {-1} & {1} \\ {0} & {-4} & {2} & {-8} \\ {2} & {0} & {3} & {-1}\end{array}\right] \quad B=\left[\begin{array}{rrrr}{1} & {3} & {-2} & {2} \\ {0} & {1} & {1} & {-5} \\ {1} & {2} & {-3} & {7} \\ {-2} & {-8} & {2} & {-1}\end{array}\right]$$Can each vector in $\mathbb{R}^{4}$ be written as a linear combination of the columns of the matrix $A$ above? Do the columns of $A$ span $\mathbb{R}^{4} ?$

only three rows contain pivot positionsnot all vectors in $\mathbb{R}^{4}$ can be written as a linear combination of columns of $\mathrm{A}$Columns of $\mathrm{A}$ do not $\operatorname{span} \mathbb{R}^{4}$Check theorem 4

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 4

The Matrix Equation Ax D b

Introduction to Matrices

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in this video, we're considering the Matrix a once again where the first thing we did with this matrix say that's provided is we reduced it to just echelon form. We found that rose 12 and three contained pivots. So we have that three of the four rows of a our pivot rose. But in other words, we could fray this phrase this by saying, Hey does not have a pivot in every row Because of this deficiency in pivots, we now have a couple statements that we can obtain immediately as direct consequences. It falls first that there exists vectors be in are for that are not leaner combinations of the columns of a If we had four pits in this matrix say that we would say all vectors in our four are linear combinations. But being deficient pivots means there are some vectors that are not such a linear combination. Well, this language here is about leaner combinations and the set of Alden your combinations is also the span of the columns of a So from the statement here, we can also say the following the columns off a do not span are for and again. That's because we required for pivots. But we found just three

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The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

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