### 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

Answered: Image /qna-images/answer/f7ab857b-4e42-41f4-b540-afdc238793bc.jpg

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...

See similar textbooks

Similar questions

A = [-2,4,-2,-4, 2,-6,-3,1, -3,8,2,-3, 0,2,5,3] Find a non-zero vector in R^4 that is an element of Nul(A) and not an element of Col(A) or explain why it's not possible to do so.
A = [-2,4,-2,-4, 2,-6,-3,1, -3,8,2,-3, 0,2,5,3] Find a non-zero vector R^4 that is an element of both Nul(A) and Col(A) or explain why it is not possible to do so.
Write each vector as a linear combination of the vectors in S. (If not possible, enter IMPOSSIBLE.) S = {(2, –1, 3), (5, 0, 4)} (a) z = (3, –4, 8) - - ( 4 )a, • ( 1 Z = (») v-(10, -) 1 59 V = 18, 4 1 7 )e, + ( V = 4 (c) w = (3, -9, 15) ( 9 s, + ( -3 W = (d) u = (4, 1, –1) u =
List five vectors in Span/v,, v2}. Do not make a sketch. -6 V1 1 V2 %3D - 5 List five vectors in Span{v1, V2} (Use the matrix template in the math palette. Use a comma to separate vect
Let v1, v2, v3 be the vectors in R^3 defined by: v1=[0 5 8], v2=[12,-20,5], v3=[-12,15,-13] a). Is {v1,v2,v3} linearly independent? Write all zeroes if it is, or if it is linearly dependent write the zero vector as a non-trivial (not all zero coefficients) linear combination of v1, v2, and v3. a). Is {v1,v2} linearly independent? Write all zeroes if it is, or if it is linearly dependent write the zero vector as a non-trivial (not all zero coefficients) linear combination of v1 and v2. c). Write the dimension of span {v1,v2,v3}
Determine if the list ( [111], [121], [011], [122] ) of vectors from vector space V = R^3 is linearly independent. If yes, show work, if no state a linear combination among the vectors of the list that adds up to a vector in the list. I am having trouble grasping the concept so if anyone can explain this problem I'd appreciate it.
lin com
Write each vector as a linear combination of the vectors in S. (Use s and s, respectively, for the vectors in the set. If not possible, enter IMPOSSIBLE.) S- ((6, -7, 8, 6), (4, 6, -4, 1)} (a) u- (-54, 127, -128, -72) -(실종 10, ). 41 75 (b) 2 4 (c) (d) z- (4, -2, 3, )
Distinguish between a row and a column vector.
The set containing the elements specified also possesses its relevant operations of multiplication over reals and vector addition. Is the set a real vector space?
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!
determine if the following pair of vectors is linearly dependent or linearly independent. show all work (3,5,6) (6,4,11) (-3,-5,6). this is vectors DO IT THIS WAY NOT THE MATRIX OTHER WAY start by making the equation like (-3,-5,6)=a (3,5,6)+ b(6,4,11)

SEE MORE QUESTIONS

Recommended textbooks for you

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

College Algebra (MindTap Course List)

Algebra

ISBN:

9781305652231

Author:

R. David Gustafson, Jeff Hughes

Publisher:

Cengage Learning

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

College Algebra (MindTap Course List)

Algebra

ISBN:

9781305652231

Author:

R. David Gustafson, Jeff Hughes

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS

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