Let Q be an orthogonal matrix and A be a nonsingular matrix. Show that cond (QA)

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Let Q be an orthogonal matrix and A be a nonsingular matrix. Show that cond (QA) = cond(A). Show that no two vectors in R3 can span all of R3. Find the subspace spanned by the three vectors [2 3 1]T, [2 1 - 5]T, and [2 4 4]T Show that the set U in Example 1.30 is not a vector subspace of V. Let V be a vector space and S = {v1,..., vk} be an arbitrary subset of V. Show that the set of all linear combinations of vectors from S is a subspace of V. Let B = {vi,..., vn} be a basis of vector space V. Show that any subset of V containing more than n vectors is linearly dependent (not linearly independent). Let u and v be arbitrary linearly independent vectors in Rn. Show that for some values of the scalars c1 and c2. the vector x = c1u + c2v has both positive and negative components. Show that the set { f1 = 2x2 + 1, f2= x2 +4x, f3 = x2 - 4x + 1 } is linearly dependent. Show that if the columns of a matrix Am Times n are linearly independent, then the matrix AT A has an inverse. Let S be a subspace of a vector space V. Show that (S ) = S. Let B1 = {u1,,.... un} and B2 = {w1,,..., wn} be two orthonormal bases of Rn, and let a1,... an and b1,..., bn be the coordinates of a vector x Rn in those bases, respectively. Show that a + .... + a = b + .... +b A = [ ] Find dim col(A) and dim N(A).

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