Linear Algebra

1. The dimension of the column space of A is the rank of A.

True

False

2. A null space is a vector space.

True

False

3. If a set of p vectors spans a p-dimensional subspace H of Rⁿ, then these vectors forma basis for H.

True

False

4. If H is a p-dimensional subspace of Rⁿ, then a linearly independent set of p vectors in H must form a basis for H.

True

False

5. If A or B is singular then AB is singular as well. (Use properties of the determinant to decide)

True

False

6. If W is a subspace of R² then W must contain the vector (0,0).

True

False

7. The dimension of the column space of A is the number of columns in RREF(A) that contain leading ones.

True

False

8. If one row of a square matrix is a multiple of another row, then the determinant of that square matrix is zero.

True

False

9. For a nonzero scalar $c$ , the vector $c \vec{v}$ is $c$ times a s long as $\vec{v}$ and has the same direction as

True

False

10. If 0 is an eigenvalue of a matrix $A$ then $A^2$ is singular.

True

False

11. If $\vec{x}$ is an eigenvector of $A$ with eigenvalue $\lambda$ then any multiple of $\vec{x}$ is also an eigenvector of $A$ with eigenvalue $\lambda$ .

True

False

12. If {v₁,..., v₄} is a linearly independent set of vectors in R⁴ then {v₁,v₂,v₃} is also linearly independent.

True

False

13. You can show that a subset of Rⁿ is not a subspace of Rⁿ by giving specific numeric examples for which the vectors do not sum to a vector in the space or for which the negative of the vector is not in the space.

True

False

14. The matrix $\begin{bmatrix} 1 & 0 & 4 \\ 0 & 6 & 3 \\ 2 & -1 & 4 \end{bmatrix}$ is non-singular.

True

False

15. If $A$ is a square matrix and $A\vec{x} = \lambda \vec{x}$ for some nonzero vector $\vec{x}$ then $\vec{x}$ is an eigenvector of $A$ .

True

False

16. If the characteristic polynomial of a matrix $A$ is $p(\lambda)=\lambda^2 +1$ then $A$ is invertible.

True

False

17. If u and v are linearly independent, and if w is in Span{u,v} then {u,v,w} is linearly dependent.

True

False

18. The vector (10, 30, -13, 14, -7, 27) can be written as a linear combination of the vectors (1,2,-3,4,-1,2), (1,-2,1,-1,2,1), (0, 2, -1, 2, -1, -1), (1,0,3,-4,1,2), and (1, -2, 1, -1, 2, -3).

True

False

19. If v₁ and v₂ are vectors in R⁴ and v₂ is not a scalar multiple of v₁ then {v₁,v₂} is linearly independent.

True

False

20. If three vectors in R³ lie in the same plane in R³, then they are linearly dependent.

True

False

21. The column space of an $m \times n$ matrix is in R^m.

True

False

22. If the determinant of the $n \times n$ matrix A is 6 then the homogeneous linear system with A as its coefficient matrix has only the trivial (all zeros) solution.

True

False

23. If a set of vectors from Rⁿ contains fewer than n vectors, then the set is linearly independent.

True

False

24. Subspace? {(u₁, u₂, u₃, u₄) in R⁴ such that 3u₁+2u₂=0}

True

False

25. Subspace? {(2s, 2s+4t, -t) in R³ such that s and t are real numbers}

True

False

26. The dimension of the column space of A and the null space of A add up to the number of rows in A.

True

False

27. Subspace? {(u₁, u₂, u₃) in R³ such that u₁ is equal to u₂}

True

False

28. If v₁, v₂, v₃ are in R³ and v₃ is not a linear combination of v₁, v₂, then {v₁,v₂,v₃} is linearly independent.

True

False

29. Subspace? {(u₁,u₂) in R² such that u₁u₂=0}

True

False

30. Subspace? {(u₁, u₂, u₃) in R³ such that u₁ is greater than u₂}

True

False

31. In general the determinant of the sum of two matrices equals the sum of the determinants of the matrix.

True

False

32. Subspace? {(u₁,u₂) in R² such that u₁²+u₂² $\quad$ is less than or equal to 1}

True

False

33. The dimension of the null space of A is the number of variables (x_i) in the equation Ax=0.

True

False

34. You can show that a subset of Rⁿ is a subspace of Rⁿ by giving specific numeric examples for which the vectors sum to a vector in the space and for which the negative of the vector is in the space.

True

False

35. If $\lambda$ is an eigenvalue of a matrix $A$ , then the linear system $(\lambda I - A) \vec{x} =\vec{0}$ has only the trivial solution.

True

False

36. The following linear system has a unique solution. $\begin{eqnarray} x_1 + x_2 -x_3 &+& 4 \\ 2x_1 -x_2+x_3 &=& 6 \\ 3x_1 -2x_2 +2x_3 &=& 0 \end{eqnarray}$

True

False

37. If W and U are subspaces of R² then the union (collection of all vectors in either W or U or both) is also a subspace of R².

True

False

38. Subspace? {(2s-2, 2s+4t, -t) in R³ such that s and t are real numbers}

True

False