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Find a basis for each of the spaces V in Exercises 16 through 36, and determine its dimension.
32. The space of all
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Linear Algebra With Applications
- Find an orthogonal matrix P such that PTAP diagonalizes the symmetric matrix A=[1331].arrow_forwardLet A and B be square matrices of order n satisfying, Ax=Bx for all x in all Rn. a Find the rank and nullity of AB. b Show that matrices A and B must be identical.arrow_forwardLet A and B be square matrices of order n over Prove or disprove that the product AB is a diagonal matrix of order n over if B is a diagonal matrix.arrow_forward
- Find two nonzero matrices A and B such that AB=BA.arrow_forwardConsider the matrices below. X=[1201],Y=[1032],Z=[3412],W=[3241] Find scalars a,b, and c such that W=aX+bY+cZ. Show that there do not exist scalars a and b such that Z=aX+bY. Show that if aX+bY+cZ=0, then a=b=c=0.arrow_forwardLabel each of the following statements as either true or false. Let A be a square matrix of order n over R such that A23A+In=On. Then A1=3InA.arrow_forward
- Take this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. Determine whether the columns of matrix A span R4. A=[1210130200111001]arrow_forwardLet A,D, and P be nn matrices satisfying AP=PD. Assume that P is nonsingular and solve this for A. Must it be true that A=D?arrow_forwardFind conditions on w,x,y,andz such that AB=BA for the matrices below. A=wxyz and B=11-11arrow_forward
- A square matrix A=[aij]n with aij=0 for all ij is called upper triangular. Prove or disprove each of the following statements. The set of all upper triangular matrices is closed with respect to matrix addition in Mn(). The set of all upper triangular matrices is closed with respect to matrix multiplication in Mn(). If A and B are square and the product AB is upper triangular, then at least one of A or B is upper triangular.arrow_forwardA square matrix is called upper triangular if all of the entries below the main diagonal are zero. Thus, the form of an upper triangular matrix is where the entries marked * are arbitrary. A more formal definition of such a matrix . 29. Prove that the product of two upper triangular matrices is upper triangular.arrow_forward
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