Suppose **a₁**, **a₂**, **a₃**, **a₄**, and **a₅** are vectors in ℝ³. Let $ A = (\mathbf{a₁} \mid \mathbf{a₂} \mid \mathbf{a₃} \mid \mathbf{a₄} \mid \mathbf{a₅}) $, and\[\text{rref}(A) =\begin{bmatrix}1 & 0 & 0 & -5 & 4 \\0 & 1 & 0 & -2 & 2 \\0 & 0 & 1 & 1 & 1\end{bmatrix}\]### Explanation of the Diagram:The matrix shown is in reduced row echelon form (rref), which is a form of a matrix used to solve linear equations. Each row corresponds to a linear equation, and the matrix representation helps identify the relationships and dependencies among the vectors $ \mathbf{a₁}, \mathbf{a₂}, \mathbf{a₃}, \mathbf{a₄}, $ and $ \mathbf{a₅} $.- The first row indicates $ 1x₁ + 0x₂ + 0x₃ = -5x₄ + 4x₅ $.- The second row shows $ 0x₁ + 1x₂ + 0x₃ = -2x₄ + 2x₅ $.- The third row presents $ 0x₁ + 0x₂ + 1x₃ = 1x₄ + 1x₅ $.The zeros in each row highlight the pivots, and the non-zero entries on the right specify relationships between the respective vectors $ \mathbf{a₄} $ and $ \mathbf{a₅} $. d. Find a basis for the column space of $ A $. If necessary, enter $\mathbf{a_1}$ for $\mathbf{a_1}$, etc., or enter coordinate vectors of the form $\langle 1, 2, 3 \rangle$ or $\langle 1, 2, 3, 4, 5 \rangle$. Enter your answer as a comma-separated list of vectors.A basis for the column space of $ A $ is $\{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\}$.e. The dimension of the null space of $ A $ is $\_\_\_\_\_$, and the null space of $ A $ is a subspace of $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$. f. If $\mathbf{x_1} = \langle 5, 2, -1, 1, 0 \rangle$, then $A\mathbf{x_1} = \_\_\_\_\_\_\_\_\_\_\_\_\_$. Is $\mathbf{x_1}$ in the null space of $ A $? $\text{choose}$g. If $\mathbf{x_2} = \langle -4, -2, -1, 0, 1 \rangle$, then $A\mathbf{x_2} = \_\_\_\_\_\_\_\_\_\_\_\_\_$. Is $\mathbf{x_2}$ in the null space of $ A $? $\text{choose}$h. If $\mathbf{x_3} = 3\mathbf{x_2} - 4\mathbf{x_1} = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_,$ then $A\mathbf{x_3} = \_\_\_\_\_\_\_\_\_\_\_\_\_$. Is $\mathbf{x_3}$ in the null space of $ A $? $\text{choose}$i. Find a basis for the null space of $ A $. If necessary, enter $\mathbf{a_1}$ for \(\

Given: The row reduction equivalence form of the matrix A is 100-54010-2200111. Explanation: d. The…

d. Find a basis for the column space of of A. If necessary, enter a1 for a, etc., or enter coordinate vectors of the form <1,2,3> or <1,2,3,4,5>. Enter your answer as a comma separated list of vectors. A basis for the column space of A is { } e. The dimension of the null space of A is , and the null space of A is a subspace of f. If x1 = (5, 2, –1, 1,0), then Ax1 = .Is x1 in the null space of A? choose v g. If x2 = (-4, –2, –1,0, 1), then Ax2 = Is x2 in the null space of A? choose v h. If xg = 3x2 – 4x1 = , then Axg = Is xg in the null space of A? choose v

d. Find a basis for the column space of of A. If necessary, enter a1 for a, etc., or enter coordinate vectors of the form <1,2,3> or <1,2,3,4,5>. Enter your answer as a comma separated list of vectors. A basis for the column space of A is { } e. The dimension of the null space of A is , and the null space of A is a subspace of f. If x1 = (5, 2, –1, 1,0), then Ax1 = .Is x1 in the null space of A? choose v g. If x2 = (-4, –2, –1,0, 1), then Ax2 = Is x2 in the null space of A? choose v h. If xg = 3x2 – 4x1 = , then Axg = Is xg in the null space of A? choose v

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

Suppose **a₁**, **a₂**, **a₃**, **a₄**, and **a₅** are vectors in ℝ³. Let $ A = (\mathbf{a₁} \mid \mathbf{a₂} \mid \mathbf{a₃} \mid \mathbf{a₄} \mid \mathbf{a₅}) $, and

\[
\text{rref}(A) =
\begin{bmatrix}
1 & 0 & 0 & -5 & 4 \\
0 & 1 & 0 & -2 & 2 \\
0 & 0 & 1 & 1 & 1
\end{bmatrix}
\]

### Explanation of the Diagram:

The matrix shown is in reduced row echelon form (rref), which is a form of a matrix used to solve linear equations. Each row corresponds to a linear equation, and the matrix representation helps identify the relationships and dependencies among the vectors $ \mathbf{a₁}, \mathbf{a₂}, \mathbf{a₃}, \mathbf{a₄}, $ and $ \mathbf{a₅} $.

- The first row indicates $ 1x₁ + 0x₂ + 0x₃ = -5x₄ + 4x₅ $.
- The second row shows $ 0x₁ + 1x₂ + 0x₃ = -2x₄ + 2x₅ $.
- The third row presents $ 0x₁ + 0x₂ + 1x₃ = 1x₄ + 1x₅ $.

The zeros in each row highlight the pivots, and the non-zero entries on the right specify relationships between the respective vectors $ \mathbf{a₄} $ and $ \mathbf{a₅} $.

d. Find a basis for the column space of $ A $. If necessary, enter $\mathbf{a_1}$ for $\mathbf{a_1}$, etc., or enter coordinate vectors of the form $\langle 1, 2, 3 \rangle$ or $\langle 1, 2, 3, 4, 5 \rangle$. Enter your answer as a comma-separated list of vectors.

A basis for the column space of $ A $ is $\{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\}$.

e. The dimension of the null space of $ A $ is $\_\_\_\_\_$, and the null space of $ A $ is a subspace of $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$.

f. If $\mathbf{x_1} = \langle 5, 2, -1, 1, 0 \rangle$, then $A\mathbf{x_1} = \_\_\_\_\_\_\_\_\_\_\_\_\_$. Is $\mathbf{x_1}$ in the null space of $ A $? $\text{choose}$

g. If $\mathbf{x_2} = \langle -4, -2, -1, 0, 1 \rangle$, then $A\mathbf{x_2} = \_\_\_\_\_\_\_\_\_\_\_\_\_$. Is $\mathbf{x_2}$ in the null space of $ A $? $\text{choose}$

h. If $\mathbf{x_3} = 3\mathbf{x_2} - 4\mathbf{x_1} = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_,$ then $A\mathbf{x_3} = \_\_\_\_\_\_\_\_\_\_\_\_\_$. Is $\mathbf{x_3}$ in the null space of $ A $? $\text{choose}$

i. Find a basis for the null space of $ A $. If necessary, enter $\mathbf{a_1}$ for \(\

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