Answer to Basic Linear Algebra Questions (1) « Mathematics in HKCEE, HKAL, and Beyond

Answer to Basic Linear Algebra Questions (1)

二月 26, 2009

The following is a list of basic questions you should know how to answer (by heart)

(1) Suppose $v_1, v_2, \dots,v_n$ are linearly independent and
$a_1v_1 + \dots + a_nv_n = 0.$ What do you know about the $a_i?$

Ans: $a_i = 0$ for all $i.$

(2) If $x \in \ker{A}$ , what is $Ax?$

Ans: Ax = 0.

(3) If $x \in Im{A}$ , then $x = ?$

Ans: x = Ay for some y.

(4) If $Ax = 0,$ then what do you know about $x?$ If $x \not= 0$ , then what do you know about $A \in M_n?$ .

Ans: x is in ker(A), if x is non-zero, then A is singular or not invertible or det(A) = 0 or…

(5) What is the necessary and sufficient condition on $k$ for $\det(A - kI) = 0 ?$

Ans: k is an eigenvalue of A.

(6) If $\{v_1, \dots, v_k\}$ is a set of vectors that span $U$ , what do you know about the dimension of $U$ ?

Ans: dim(U) <= k.

(7) Can you give a set of vectors that span $U$ but are not linearly independent?

Ans: Yes, take the original spanning set $\{v_1, \dots, v_k\}$ and throw in the zero vector to obtain $\{ v_1, \dots, v_k, 0\}.$ Note that any set containing the zero vector must be linear dependent. Also, since you are expanding the set of vectors, this set still span U.

(8) Can you give a set of linearly independent vectors in $R^n$ but does not span $R^n$ ? If so, for what $n$ ? Or all $n$ ?

Ans: Yes, if n > 1, then simply take the set to be $\{ v \}$ where v is ANY non-zero vector in R^n. If n = 1, it is NOT possible to find such a set because any non-zero vector will be a basis for R, while any set containing the zero vector must be linearly dependent.

(9) What does it mean for a matrix $A$ being diagonalizable? Do you know any theorems that tell you when you can diagonalize a matrix $A$ ?

Ans: It means A is similar to a diagonal matrix D, or more explicitly, it means that there exists an invertible matrix P such that $P^{-1}AP = D.$

(a) A is diagonalizable iff the algebraic and geometric multiplicity for each eigenvalue of A coincide.

(b) A is diagonalizable iff the minimal polynomial of A has no repeated roots.

(10) What does it mean for two matrices being similar? In general, how can you tell whether two matrices are similar? Are there quick ways to tell whether two matrices are NOT similar?

Ans: A and B are similar if there exists invertible matrix Q such that $Q^{-1}AQ = B.$ The only way to tell whether two matrices are similar is by show that they have the SAME Jordan canonical form. (which is unique up to permutation of the individual Jordan Blocks)

Yes, there are many properties that are preserved by similarity, if any of these preserved properties is different for A and B, then A is NOT similar to B. For example, if tr(A) is NOT equal to tr(B), then A and B are NOT similar.

Warning: Note that even if tr, det, rank, alg mult, geo mult, charpoly…etc are all the same for A and B, this DOES NOT mean A and B are similar. The ONLY way to tell whether A and B are similar is to show that they have the same Jordan Form. (at least this is the only I know)

(11) Suppose $A$ is nilpotent. Prove that $A$ is singular in as many ways as you can.

Ans: Exercise.

(12) If $\{v_1, v_2, \dots, v_n\}$ is a basis for $V$ , and $a_1v_1 + \dots + a_nv_n = 0.$ What do you know about $a_i?$ If $v \in V$ , what do you know about $v$ in terms of $v_1, \dots, v_n?$

Ans: $a_i = 0$ since part of the criterion of being a basis is lin. indenp. If $v \in V$ , we may express v as a linear combination of the basis vector, i.e. $v = a_1v_1 + \dots + a_nv_n$ for some $a_i.$

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Mathematics in HKCEE, HKAL, and Beyond

Answer to Basic Linear Algebra Questions (1)

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