20. Let T: R¹ R³ be the linear transformation T X1 X2 X3 X4 x1 + x2 x3 + x4 (x₁+x2+x3+x4, Compute the matrix for T with

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20 Let T R R Be The Linear Transformation T X1 X2 X3 X4 X1 X2 X3 X4 X X2 X3 X4 Compute The Matrix For T With 1 (62.74 KiB) Viewed 70 times

20. Let T: R¹ R³ be the linear transformation T X1 X2 X3 X4 x1 + x2 x3 + x4 (x₁+x2+x3+x4, Compute the matrix for T with respect to the standard bases for R* and R³. Com- pute kernel (7), nullity (7), image (T), and rank(7), and verify that nullity (7) + rank (T) = 4. 21. Let T: VW be a linear transformation with dim(V) = n. Let {fi,... f} be a linearly independent subset of kernel (T). Prove that if V EV and T (V) #0, then {V+fi...,V+fk} is a linearly independent subset of V. 22. Consider the linear transformation T: R³ → R² defined by T(x, y, z) = (x, y,0). Compute kernel (T) and nullity (T). Construct three linearly independent vectors in V\kernel (T). 23. Let V and W be finite-dimensional vector spaces and let T: VW be a linear transformation. a. Letr = rank (T) and let {f₁,...,f,} be a basis for image (T). For all i = 1,...,r, choose a vector e; € V such that T (e;) = f₁. Prove that {e₁,..., e,} is a linearly independent set of vectors in V. b. Let k = nullity (T) and let {er+1,..., er+k} be a basis for kernel (7). Prove that the set {e₁,..., er, er+1,..., er+k} is linearly independent. c. Without using Theorem 4.13, prove directly that {e₁,..., er, er+er+k} is a basis for V. This gives another proof of Theorem 4.13.