### Linear Algebra - Vector SpacesGiven the matrix \( A \) and vector \( \mathbf{b} \):\[ A = \begin{bmatrix}2 & 0 & 6 \\-1 & 8 & 5 \\1 & -2 & 1 \end{bmatrix}\]\[ \mathbf{b} = \begin{bmatrix}10 \\3 \\3 \end{bmatrix}\]Let \( W \) be the set of all linear combinations of the columns of \( A \).#### Questions to Consider:**a. Is \(\mathbf{b}\) in \(W\)?****b. Show that the third column of \( A \) is in \( W \)?**#### Explanation:1. **Vector \( \mathbf{b} \) in \( W \):** To determine if \(\mathbf{b}\) is in \( W \), we need to solve the equation \( A\mathbf{x} = \mathbf{b} \) for some vector \(\mathbf{x}\). If a solution \(\mathbf{x}\) exists, then \(\mathbf{b}\) is a linear combination of the columns of \( A \), and thus is in \( W \).2. **Third Column of \( A \) in \( W \):** The third column of \( A \) can be written as: \[ \mathbf{a}_3 = \begin{bmatrix} 6 \\ 5 \\ 1 \end{bmatrix} \] To show that \(\mathbf{a}_3\) is in \( W \), we need to express \(\mathbf{a}_3\) as a linear combination of the columns of \( A \). This means finding scalars \( c_1 \), \( c_2 \), and \( c_3 \) such that: \[ c_1 \mathbf{a}_1 + c_2 \mathbf{a}_2 + c_3 \mathbf{a}_3 = \mathbf{a}_3 \] where \( \mathbf{a}_1 \), \( \mathbf{a}_2 \), and \( \mathbf{a}_3 \) are the first, second, and third columns of

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Answered: 20 6 10 Let A = -1 8 5 and let b = B- 1…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.5: Basis And Dimension

Problem 65E: Find a basis for the vector space of all 33 diagonal matrices. What is the dimension of this vector...

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Let four vectors ~v1, ~v2, ~v3 and ~v4 be given by~v1 = (1, 0, 2), ~v2 = (0, 1, −1),~v3 = (1, 8, −6), ~v4 = (1, 2, 0); Let the vector space V be the linear span of ~v1, ~v2, ~v3 and ~v4.(a). Find all values of k such that the vector ~u = (2, 1, k) belongs to V .Explain. a.2)Are the vectors {~v1, ~v2} linearly independent? Explain. a.3)Find the dimension and a basis of the space V . a.4)Let A be the matrix whose columns are the vectors ~v1, ~v2, ~v3 and ~v4(written as columns). Find a basis and the dimension of the solutionset of the homogeneous linear system AX = 0. Please write full solution with clear work thank you!
tion 4 of 12 > Calculate the cross product. (9j - 5k) x (2j+ k) (Give your answer using standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.) ai + bj + ck = lu Transcendentals Pub
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20 6 10 Let A = -1 8 5 and let b = B- 1 -2 1 3 W be the set of all linear combinations of the columns of A. a. Is b in W? b. Show that the third column of A is in W? 3 and