LINEAR ALGEBRA TEST 3 T OR F Q&A
If A is symmetric matrix, then A is diagonalizable - Answer-True
If A and B are orthogonal nxn matrices , then so is A+B - Answer-False, for example , If
A is any orthogonal matrix , then B=-A is also orthogonal, but then A+B is the zero
matrix , which is not orthogonal.
The eigen values for a square matrix A are real - Answer-False. The eigen values for [1
1, -1,1] are 1+- i
If v1 and v2 are in a vector space V, then so is v1-v2 - Answer-True
If S1 and S2 are subspaces of a vector space V , then the intersections S1 ^ S2 is also
a subspace of V - Answer-True
If S1 and S2 are subspaces of a vector space V, then the union S1 U S2 is also a
subspace of V - Answer-False, since S1 U S2 is not closed under vector addition
If A is not invertible, then A is not diagonalizable. - Answer-False, for example, [1 0, 0 0]
is not invertible, but is diagonalizable
If A and B are diagonalizable nxn matrices, then so is AB - Answer-False. Consider A=
[1 1, 0 0] and B=[ -1 2, 1 0]
If A and B are diagonalizable nxn matrices, then so is A+B - Answer-False. Consider
A=[1 1, 0 -1] and B=[-1 1, 0 1]
If A is a rotation-dilation matrix, then the angle between a vector Xo and AXo is the
same for all nonzero Xo in R^2 - Answer-True
The amount of dilation imparted by a rotation-dilation matrix A is equal to I lambda I,
where lambda is and eigenvalue of A - Answer-True
If A=[a -b, b a] is a rotation-dilation matrix that has the eigenvalues lambda1 and lambda
2, then I lambda1 I = I lambda2 I - Answer-True
If II u-v II= 5, then the distance between 2u and 2v is 20 - Answer-False, since II 2u-2v II
= II 2 (u-v) II= I 2 I IIu-vII= 2(5)=10
If u1 dot u2=0 and u2 dot u3= 0, then u1 dot u3=0 - Answer-False, for example, u1=[1
0], u2=[0 1], and u3=[-1 0]
Every subspace S of R^n has an orthonormal basis - Answer-True
If A is symmetric matrix, then A is diagonalizable - Answer-True
If A and B are orthogonal nxn matrices , then so is A+B - Answer-False, for example , If
A is any orthogonal matrix , then B=-A is also orthogonal, but then A+B is the zero
matrix , which is not orthogonal.
The eigen values for a square matrix A are real - Answer-False. The eigen values for [1
1, -1,1] are 1+- i
If v1 and v2 are in a vector space V, then so is v1-v2 - Answer-True
If S1 and S2 are subspaces of a vector space V , then the intersections S1 ^ S2 is also
a subspace of V - Answer-True
If S1 and S2 are subspaces of a vector space V, then the union S1 U S2 is also a
subspace of V - Answer-False, since S1 U S2 is not closed under vector addition
If A is not invertible, then A is not diagonalizable. - Answer-False, for example, [1 0, 0 0]
is not invertible, but is diagonalizable
If A and B are diagonalizable nxn matrices, then so is AB - Answer-False. Consider A=
[1 1, 0 0] and B=[ -1 2, 1 0]
If A and B are diagonalizable nxn matrices, then so is A+B - Answer-False. Consider
A=[1 1, 0 -1] and B=[-1 1, 0 1]
If A is a rotation-dilation matrix, then the angle between a vector Xo and AXo is the
same for all nonzero Xo in R^2 - Answer-True
The amount of dilation imparted by a rotation-dilation matrix A is equal to I lambda I,
where lambda is and eigenvalue of A - Answer-True
If A=[a -b, b a] is a rotation-dilation matrix that has the eigenvalues lambda1 and lambda
2, then I lambda1 I = I lambda2 I - Answer-True
If II u-v II= 5, then the distance between 2u and 2v is 20 - Answer-False, since II 2u-2v II
= II 2 (u-v) II= I 2 I IIu-vII= 2(5)=10
If u1 dot u2=0 and u2 dot u3= 0, then u1 dot u3=0 - Answer-False, for example, u1=[1
0], u2=[0 1], and u3=[-1 0]
Every subspace S of R^n has an orthonormal basis - Answer-True
