On 2.4 number 16 show all work In Exercises 13-15, find a basis for the subspace spannedby the given vectors. (These do not have unique an-2.4Dimension; Nullspace, Row Space, and Column Space105b) Find a subset of pi(x), p2(x), p3(x), pa(x) that-1forms a basis for P2 in the manner of the proofof part (2) of Lemma 2.11.22. If the initial points of vectors in R' are placed at11.vectors.3.3the origin, what geometric object is a subspace ofR of dimension one? What geometric object is a-1-12 0subspace of R of dimension two?23. Explain why the reduced row-echelon form of a ma-11-1 0-212.-21trix is unique. Is the reduced column-echelon formunique? Why or why not?24. Use the system of linear equation solving capabili-ties of Maple or another appropriate software pack-age to show that the matricesthe givenswers.)5-613.1-33-7-3 -331311-216 -2117-19114.-3 466 -9832 -9-1-2 -12215 -5-4 015.form a basis for M3×2(R).25. Let A be the matrix16. Without formally showing that there is a nontriviallinear combination of x4-2x, x+x³ – 1, x³+x+3,x - x² – x4, 10x – 91, ax4 + V3x³ - 7x2 equalto the zero polynomial, we can conclude these poly-nomials are linearly dependent. Why is this?-23-14-2 73-18A =17 -5 112:0 12 -5 15 817. Show that det(A) +0 for an n x n matrix A if andonly if rank(A) = n.a) In Maple, the command nullspace (orequivalently, kernel) can be used to find a basisfor the nullspace of a matrix. (The basis vectorswill be given as row vectors instead of columnvectors.) Use this command or a correspondingcommand in an appropriate software package tofind a basis for the nullspace of the matrix A.K.18. Suppose that A is an m x n matrix. Show that ifm n, then the rows of A are linearly dependent.atrix.19. Suppose that A is an m x n matrix. Show that ifm

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On 2.4 number 16 show all work In Exercises 13-15, find a basis for the subspace spannedby the given vectors. (These do not have unique an-2.4Dimension; Nullspace, Row Space, and Column Space105b) Find a subset of pi(x), p2(x), p3(x), pa(x) that-1forms a basis for P2 in the manner of the proofof part (2) of Lemma 2.11.22. If the initial points of vectors in R' are placed at11.vectors.3.3the origin, what geometric object is a subspace ofR of dimension one? What geometric object is a-1-12 0subspace of R of dimension two?23. Explain why the reduced row-echelon form of a ma-11-1 0-212.-21trix is unique. Is the reduced column-echelon formunique? Why or why not?24. Use the system of linear equation solving capabili-ties of Maple or another appropriate software pack-age to show that the matricesthe givenswers.)5-613.1-33-7-3 -331311-216 -2117-19114.-3 466 -9832 -9-1-2 -12215 -5-4 015.form a basis for M3×2(R).25. Let A be the matrix16. Without formally showing that there is a nontriviallinear combination of x4-2x, x+x³ – 1, x³+x+3,x - x² – x4, 10x – 91, ax4 + V3x³ - 7x2 equalto the zero polynomial, we can conclude these poly-nomials are linearly dependent. Why is this?-23-14-2 73-18A =17 -5 112:0 12 -5 15 817. Show that det(A) +0 for an n x n matrix A if andonly if rank(A) = n.a) In Maple, the command nullspace (orequivalently, kernel) can be used to find a basisfor the nullspace of a matrix. (The basis vectorswill be given as row vectors instead of columnvectors.) Use this command or a correspondingcommand in an appropriate software package tofind a basis for the nullspace of the matrix A.K.18. Suppose that A is an m x n matrix. Show that ifm > n, then the rows of A are linearly dependent.atrix.19. Suppose that A is an m x n matrix. Show that ifm < n, then the columns of A are linearly depen-dent.b) Bases for the row space of a matrix can be foundwith Maple by using the gausselim or gaussjord(or equivalently, rref) commands. Such basesmay also be found using the rowspace androwspan commands. Find bases for row spaceof the matrix A using each of these four Maplecommands or corresponding commands inanother appropriate software package. If you20. Consider the linearly independent polynomialsP1(x) =X* + x, p2(x) = x + 1. Find a poly-Lemma 2.11.span P2.2 21

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-4 -11 25 16. Without formally showing that there is a nontrivial linear combination of x-2x, x++x³ -1,x³+x+3, x-x2 - x, 10x - 91, 7x+ 3x- 7x2 equal to the zero polynomial, we can conclude these poly- nomials are linearly dependent. Why is this? if and