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Chapter 3 Solutions
Linear Algebra: A Modern Introduction
- Find a basis B for R3 such that the matrix for the linear transformation T:R3R3, T(x,y,z)=(2x2z,2y2z,3x3z), relative to B is diagonal.arrow_forwardIn Exercises 7-10, find the standard matrix for the linear transformation T. T(x,y)=(3x+2y,2yx)arrow_forwardLet T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.arrow_forward
- Let 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_forwardExplain what it means in terms of an inverse for a matrix to have a 0 determinant.arrow_forwardLet TA: 23 be the matrix transformation corresponding to A=[311124]. Find TA(u) and TA(v), where u=[12] v=[32].arrow_forward
- True or false? det(A) is defined only for a square matrix A.arrow_forwardLet T be a linear transformation from R2 into R2 such that T(1,0)=(1,1) and T(0,1)=(1,1). Find T(1,4) and T(2,1).arrow_forwardIn Exercises 1-12, determine whether T is a linear transformation. T:MnnMnn defines by T(A)=AB, where B is a fixed nn matrixarrow_forward
- In Exercises 20-25, find the standard matrix of the given linear transformation from ℝ2 to ℝ2. 24. Reflection in the line y = xarrow_forwardFind an orthogonal matrix P such that PTAP diagonalizes the symmetric matrix A=[1331].arrow_forwardIn Exercises 20-25, find the standard matrix of the given linear transformation from2to 2. Projection onto the line y=2xarrow_forward
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