(4) Let A = [a1 a2 a3 a4] be a matrix whose columns a1, a2, a3, and a4 are elements of R4. It is known that a4 = a1+a3. What is a nontrivial solution to the system Ax = 0? SHORT ANSWER - NO EXPLANATION REQUIRED (5) Let A be a 3 x 2 matrix with 2 pivot positions. [1 point each] (a) Is the system Ax = b consistent for all b in R³? (b) Do the columns of A span R³? (c) Is the system Ax = 0 consistent? (d) How many free variables does A have? (6) Let A be a 2 x 3 matrix with 2 pivot positions. [1 point each] (a) Is the system Ax = b consistent for all b in R²? (b) How many different linear combinations of the columns of A exist? (c) Do the columns of A span R²? (d) How many basic variables does A have? (7) Let A be a 3 x 3 matrix with 2 pivot positions. [1 point each] (a) Does the system Ax = 0 have nontrivial solutions? (b) Can each b in R³ be written as a linear combination of the columns of A? (c) Can any of the columns of A be written as a linear combination of the other columns? (d) Is the system Ax = b consistent for all b in R³?
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Since a4 = a1 + a3, we can rewrite the matrix A as: A = [a1 a2 a3 (a1 + a3)] Now, let's consider the system Ax = 0: [a1 a2 a3 (a1 + a3)] [x1] = 0 [x2] [x3] [x4] We can rewrite this system as: Show more…
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Let a1, a2, ..., a5 denote the columns of the matrix A shown to the right and let B = [ a1 a2 a4 ]. Answer parts (a) through (c) below. a. Explain why a3 and a5 are in the column space of B. Choose the correct answer below. A. The column space of B is all of R4. B. The systems Bx = a3 and Bx = a5 are both inconsistent. C. The reduced row echelon forms for [B a3] and [B a5] are the same. D. The free variable columns can always be written as a linear combination of the pivot columns. E. The pivot columns can always be written as a linear combination of the free variable columns. b. Find a set of vectors that spans Nul A. (Use a comma to separate vectors as needed.)
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Let a1, a2,..., a5 denote the columns of the matrix A shown to the right and let B = [ a1 a2 a4 ]. Answer parts (a) through (c) below. a. Explain why a3 and a5 are in the column space of B. Choose the correct answer below. A. The pivot columns can always be written as a linear combination of the free variable columns. B. The reduced row echelon forms for [B a3] and [B a5] are the same. C. The free variable columns can always be written as a linear combination of the pivot columns. D. The systems Bx = a3 and Bx = a5 are both inconsistent. E. The column space of B is all of R4. b. Find a set of vectors that spans Nul A. A spanning set for Nul A is { [ -1/3 -1/3 1 0 0 ], [ -11/3 26/3 0 5 1 ] }. c. Let T : R5->R4 be defined by T(x) = Ax. Explain why T is neither one-to-one nor onto. Choose the correct answer below. A. The vector spaces R5 and R4 have different dimensions. B. The reduced row echelon form of A shows that the columns of A are linearly dependent and do not span R4. C. The reduced row echelon form of A shows that the columns of A are linearly dependent and span R4. D. The reduced row echelon form of A shows that the columns of A are linearly independent and do not span R4.
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