n
For any matrix A，let N(A) denote its null space. In real space R ，define the
1
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2
inner product < x，y > = x1y1 + x2y2 + ... + xnyn and 2-norm ∥x∥= < x，x >
T T
for all vectors x = [x1···xn] and y = [y1···yn] . Suppose S is a
n ┴ n
subspace of R . Let S be the orthogonal complement of S in R with respect
to the inner product <．,．>. Consider a real 4x4 matrix A with N(A) spanned
T T T
by the set { [2 0 2 -1] ， [1 2 0 -1] ， [3 -1 4 -1] }
(1). What is the rank of A ?
┴
(2). Find an orthonormal basis B for N(A)
(3). With the above information, judge which one of the following three
conditions can uniquely determine the matrix A, and find the unique A under
T T
that condition: Condition I: A = A and det(A)=0; Condition II: A = A and A
T
has an eigenvalue 1; Condition III: A = -A .
T
(4). What is the least square error solution of Ax=[1 1 1 1] for the unique
matrix A obtained in (3) ?
(1)、(2)沒問題，(3)、(4)不會做，好難唷..想不太到
煩請高手了
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