Linear Algebra: Exam-Focused Concepts and Problem Solving: 30 flashcards
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1. What is the standard form of a linear equation in $x$ and $y$?
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$ax + by = c$, where each variable appears only to the first power and no variable is multiplied by another.
2. Why is $x + y^2 = 0$ not a linear equation?
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Because of the $y^2$ term: squaring a variable is a nonlinear operation.
3. State the formula for the inverse of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$.
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$\frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$, provided $ad-bc \neq 0$.
4. When does a $2 \times 2$ matrix have no inverse?
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When its determinant $ad-bc = 0$.
5. What is an elementary matrix?
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A matrix obtained by applying exactly one elementary row operation to an identity matrix.
6. What is the result of $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}^{30}$?
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$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ (the identity), because the matrix is a reflection and its even powers are the identity.
7. State the multiplicative property of determinants.
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$\det(AB) = \det(A)\det(B)$ for square matrices of the same size.
8. For which values of $k$ is $\begin{pmatrix} k-2 & -2 \\ -2 & k-2 \end{pmatrix}$ not invertible?
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$k = 0$ and $k = 4$.
9. Compute $(0,5) + (-1,4)$.
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$(-1, 9)$.
10. If $\|v\| = 7$, what are the possible values of $k$ such that $\|kv\| = 5$?
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$k = \pm \frac{5}{7}$.
11. What does it mean for a vector $w$ to be a linear combination of $u$ and $v$?
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There exist scalars $a$ and $b$ such that $w = au + bv$.
12. How can you test whether two vectors in $\mathbb{R}^2$ form a basis?
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Form the matrix with the vectors as columns; they form a basis if and only if its determinant is nonzero.
13. What is a Hermitian matrix?
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A square complex matrix $A$ satisfying $A^* = A$, where $A^*$ is the conjugate transpose.
14. What is a skew-Hermitian matrix?
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A square complex matrix $A$ satisfying $A^* = -A$.
15. For $u = (2-i, 4i, 1+i)$, what is $\|u\|$?
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$\sqrt{23}$.
16. List the ten axioms of a vector space.
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Closure under addition and scalar multiplication; commutativity and associativity of addition; existence of zero vector and additive inverses; four distributive/associativity properties linking scalars and vectors.
17. State the Cauchy–Schwarz inequality.
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$|\langle u,v \rangle| \leq \|u\|\|v\|$, with equality exactly when $u$ and $v$ are linearly dependent.
18. Are $u = (1,2,-1)$ and $v = (0,1,1)$ orthogonal? Why or why not?
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No, because $u \cdot v = 1 \neq 0$.
19. What is the first step in Gaussian elimination?
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Write the augmented matrix of the system.
20. How do you find the inverse of a matrix using row operations?
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Augment $A$ with the identity matrix $I$ and row-reduce until the left block becomes $I$; the right block is then $A^{-1}$.
21. What are the eigenvalues of $\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 4 & -17 & 8 \end{pmatrix}$?
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$4$, $2+\sqrt{3}$, and $2-\sqrt{3}$.
22. Why is the set $\{(a,1,1) : a \in \mathbb{R}\}$ not a subspace of $\mathbb{R}^3$?
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It does not contain the zero vector $(0,0,0)$.
23. How can you express a polynomial as a linear combination of other polynomials?
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Equate coefficients of like powers, set up a linear system, and solve for the scalars.
24. What is the determinant test for three vectors in $\mathbb{R}^3$ to form a basis?
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Form the matrix with the vectors as columns; they form a basis if and only if its determinant is nonzero.
25. What does it mean to diagonalize a matrix?
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To find an invertible matrix $P$ and a diagonal matrix $D$ such that $A = PDP^{-1}$, using a full set of eigenvectors as columns of $P$.
26. What is the Gram–Schmidt process used for?
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To orthogonalize a basis by subtracting projections onto earlier vectors.
27. What is the quickest way to rule out a candidate subspace?
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Check whether it contains the zero vector; if not, it is not a subspace.
28. State the distributive property of scalar multiplication over scalar addition in a vector space.
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$(k_1+k_2)v = k_1v + k_2v$.
29. What is the determinant of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?
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$ad - bc$.
30. If $A$ is invertible, what is $\det(A^{-1})$ in terms of $\det(A)$?
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$\det(A^{-1}) = \frac{1}{\det(A)}$.
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