Consider the matrixA=⎡⎣⎢132264−1−3−2396⎤⎦⎥.Find a basis for the null space of A. A.⎧⎩⎨⎪⎪⎪⎪⎪⎪⎡⎣⎢⎢⎢0−200⎤⎦⎥⎥⎥,⎡⎣⎢⎢⎢0010⎤⎦⎥⎥⎥,⎡⎣⎢⎢⎢000−3⎤⎦⎥⎥⎥⎫⎭⎬⎪⎪⎪⎪⎪⎪ B.⎧⎩⎨⎪⎪⎪⎪⎪⎪⎡⎣⎢⎢⎢−2100⎤⎦⎥⎥⎥,⎡⎣⎢⎢⎢1010⎤⎦⎥⎥⎥,⎡⎣⎢⎢⎢−3001⎤⎦⎥⎥⎥⎫⎭⎬⎪⎪⎪⎪⎪⎪ C.⎧⎩⎨⎪⎪⎪⎪⎪⎪⎡⎣⎢⎢⎢0000⎤⎦⎥⎥⎥,⎡⎣⎢⎢⎢1−213⎤⎦⎥⎥⎥⎫⎭⎬⎪⎪⎪⎪⎪⎪ D.⎧⎩⎨⎪⎪⎪⎪⎪⎪⎡⎣⎢⎢⎢12−13⎤⎦⎥⎥⎥⎫⎭⎬⎪⎪⎪⎪⎪⎪

Find the basis for the kernel of the matrixA=⎛⎝⎜11−12−105−41001⎞⎠⎟A. {⎛⎝⎜⎜⎜1−311⎞⎠⎟⎟⎟} B. {⎛⎝⎜⎜⎜1−310⎞⎠⎟⎟⎟}C. {⎛⎝⎜⎜⎜1310⎞⎠⎟⎟⎟} D. {⎛⎝⎜⎜⎜−1310⎞⎠⎟⎟⎟}

Find a basis for the column space ofA=⎡⎣⎢⎢−43820−4−2−34010⎤⎦⎥⎥.

Use the fact that matrices A and B are row-equivalent.A = −2 −5 8 0 −17 1 3 −5 1 5−5 −9 13 7 −671 7 −13 5 −3B = 1 0 1 0 1 0 1 −2 0 30 0 0 1 −50 0 0 0 0(a) Find the rank and nullity of A.rank nullity     (b) Find a basis for the nullspace of A. (c) Find a basis for the row space of A. (d) Find a basis for the column space of A.

If 𝑆 = {𝑣1 , 𝑣2, . . . , 𝑣𝑛 } is a set of vectors in a finite dimensional vector space 𝑉 , then 𝑆 is called a basis for 𝑉 if:

A𝐴 is a 5×55×5 matrix with columns given by the vectors a1,a2,a3,a4,a5𝑎1,𝑎2,𝑎3,𝑎4,𝑎5  so,A=(a1  a2  a3  a4  a5)𝐴=(𝑎1  𝑎2  𝑎3  𝑎4  𝑎5) If you need to refer to these column vectors in any answer use a1 for a1𝑎1 , a2 for a2𝑎2 etc. A𝐴  has row-reduced echelon form (RREF)U=𝑈= ⎛⎝⎜⎜⎜⎜⎜⎜1000020000010000010000010⎞⎠⎟⎟⎟⎟⎟⎟1200000100000100000100000 State the rank and nullitty of A𝐴 .rank(A)=rank(𝐴)=   nullity(A)=nullity(𝐴)=     Find a basis for the kernel or nullspace of  A𝐴 ,  ker(A)ker(𝐴) . Your answer must be a set of vectors, that is a sequence of vectors separated by commas enclosed in curly brackets { }.  Vectors can either be entered either using Maple notaation, for example,   < 3, 7 >  for the vector (37)(37) or in terms of the vectors   a1,a2,a3,a4,a5𝑎1,𝑎2,𝑎3,𝑎4,𝑎5  written as  a1, a2, a3, a4, a5.The syntax for typical answers is one of the following two forms: { <1,2,3>, <4,5,6>, <1,1,1> }  or  { a2, a3, a4 } Basis for  ker(A)=ker(𝐴)=