Let A be an m × n matrix with columns C₁, C₂, ..., C₁. If rank A = n, show that {A¹c₁, A¹ C₂, ..., AT cn} is a basis of R”.

Q: If         →                    AB = (−5, 4) then 4 is the _____ of the vector.

A: We know in a vector of the form (a, b); a is the x-component and b is the y-component of the vector.

Q: Explain why S is not a basis for M2,2.

A: Let's check if S is linearly independent or not. S will not be linearly independent if for some…

Q: Let M be the set of all vectors in the form  ...(see attached picture)  where a , b and c are…

A: Notice that…

Q: Explain why S is not a basis for P,. S = {1, 3x, -4 + x2, 6x} O sis linearly dependent. s does not…

A:

Q: 2. Show that the following is a basis for M, 2x2 [3 6 3 1 0 -1 2 -8 12

A:

Q: Explain why S is not a basis for P2. S = {1, 4x, -9 + x², 7x} S is linearly dependent. O s does not…

A: Given S={1,4x,-9+x2,7x}. We explain why S is not a basis for P2.

Q: -1 1 4 Let A = -3 3 12 -4 4 16 Find a basis for Null(A).

A: We are given a matrix of order 3×3 A= -1 1 4       -3 3 12       -4 4 16

Q: [2 1 1. Let A = 1 1 1 0 2 -1 | 1 Determine a basis for col(A), null(A), row(A), and null(A").

A: Given:A=210-1111102-10 We need to find : (a) Column space of matrix A (b) Null space of matrix A (c)…

Q: Determine if the statements are true or false. 1. Any four vectors in R³ are linearly dependent. 2…

A:

Q: 1 3 3. Let S 2 3 2 a. Is span(S)=R³? b. Determine whether S is the basis of R3.

A:

Q: -2 -5 8 0 -17] 1 1. Consider A = -5 1 5 11 -19 7 1 1 7 -13 5 -3 a. Find the basis for, and the…

A:

Q: Which of the following is a basis for R? O a. {(1,0), (2,0)} O b. (1,2), (2,3), (1,7)} O c.…

A: We have to check

Q: Determine whether S is a basis for P3.S = {4t − t2, 5 + t3, 5 + 3t, −3t2 + 2t3}

A: The given set is S=4t-t2, 5+t3, 5+3t, -3t2+2t3. Check whether the set S is a basis for the vector…

Q: Determine whether b is in the column space of A. [-10 21 A = 5 8-10, b = 3 [-3-36 1. A

A:

Q: We showed in the text that z; - c; = 0 for a basic variable. Interpret sult. Y leaves the basis. it

A: Answer given below.

Q: B 5. Which of the following is not a basis for R²? a. (1,2),(3,8) b. (2,2),(3,2) c. (2,3),(4,6) d.…

A:

Q: The set S = {(1, 1, 0), (-2, 4, 1), (3, 2, -5), (-1, 3, -2)} is: Select one: a. Dependent. O b.…

A:

Q: If x has a value of -4, determine if the series converges (absolutely or conditionally) or…

A:

Q: Determine if the statements are true or false. 1. Any four vectors in R3 are linearly dependent. ?…

A:

Q: Determine if the statements are true or false. 1. Any four vectors In R³ are linearly dependent. ?…

A: Since you have posted a multi subparts question according to guildlines I will solve first(1,2,3)…

Q: 9. Given B = { (3,0,-3), (3,-2,1), (-1, -6,1)} and B' = {(1,1,–1),(0,1,2),(-1,1,0)}, find the…

A:

Q: Explain why S is not a basis for P2. S = {1 – x, 1 – x², –1 – 3x + 4x²} O s is linearly dependent. O…

A:

Q: ?:R3 →R2,with? ?,?,? =(?+2?+3?,4?+?−2?) Find a basis for the image and state its dimension.

A:

Q: 1. Determine the linearly independent vectors of the following set. Show the steps for your result.…

A:

Q: -2 -5 8 -17] -5 1 1. Consider A = 1 3 11 -19 7 1 1 7 -13 5 -3 a. Find the basis for, and the…

A: