- Let H denote span. Find the dimension of H and determine a basis.
- Let M = . Is M a subspace? Explain.
- Let M = . Is M a subspace? Explain.
- Let w ∈ ℝ
^{4}and let M =. Is M a subspace? Explain. - Let M = . Is M a subspace? Explain.
- Let w,w
_{1}be given vectors in ℝ^{4}and defineIs M a subspace? Explain.

- Let M = . Is M a subspace? Explain.
- Let M = . Is M a subspace? Explain.
- Suppose is a set of vectors from F
^{n}. Show that 0 is in span. - Consider the vectors of the form
Is this set of vectors a subspace of ℝ

^{3}? If so, explain why, give a basis for the subspace and find its dimension. - Consider the vectors of the form
Is this set of vectors a subspace of ℝ

^{4}? If so, explain why, give a basis for the subspace and find its dimension. - Consider the vectors of the form
Is this set of vectors a subspace of ℝ

^{4}? If so, explain why, give a basis for the subspace and find its dimension. - Let V denote the set of functions defined on . Vector addition is defined as≡ f+ gand scalar multiplication is defined as≡ α. Verify V is a vector space. What is its dimension, finite or infinite? Justify your answer.
- Let V denote the set of polynomial functions defined on . Vector addition is defined as≡ f+ gand scalar multiplication is defined as≡ α. Verify V is a vector space. What is its dimension, finite or infinite? Justify your answer.
- Let V be the set of polynomials defined on ℝ having degree no more than 4. Give a basis for this vector space.
- Let the vectors be of the form a + bwhere a,b are rational numbers and let the field of scalars be F = ℚ, the rational numbers. Show directly this is a vector space. What is its dimension? What is a basis for this vector space?
- Let V be a vector space with field of scalars F and suppose is a basis for V . Now let W also be a vector space with field of scalars F. Let L :→ W be a function such that Lv
_{j}= w_{j}. Explain how L can be extended to a linear transformation mapping V to W in a unique way. - If you have 5 vectors in F
^{5}and the vectors are linearly independent, can it always be concluded they span F^{5}? Explain. - If you have 6 vectors in F
^{5}, is it possible they are linearly independent? Explain. - Suppose V,W are subspaces of F
^{n}. Show V ∩W defined to be all vectors which are in both V and W is a subspace also. - Suppose V and W both have dimension equal to 7 and they are subspaces of a vector space of dimension 10. What are the possibilities for the dimension of V ∩ W? Hint: Remember that a linear independent set can be extended to form a basis.
- Suppose V has dimension p and W has dimension q and they are each contained in a
subspace, U which has dimension equal to n where n > max. What are the possibilities for the dimension of V ∩W? Hint: Remember that a linear independent set can be extended to form a basis.
- If b≠0, can the solution set of Ax = b be a plane through the origin? Explain.
- Suppose a system of equations has fewer equations than variables and you have found a solution to this system of equations. Is it possible that your solution is the only one? Explain.
- Suppose a system of linear equations has a 2 × 4 augmented matrix and the last column is a pivot column. Could the system of linear equations be consistent? Explain.
- Suppose the coefficient matrix of a system of n equations with n variables has the property that every column is a pivot column. Does it follow that the system of equations must have a solution? If so, must the solution be unique? Explain.
- Suppose there is a unique solution to a system of linear equations. What must be true of the pivot columns in the augmented matrix.
- State whether each of the following sets of data are possible for the matrix equation Ax = b.
If possible, describe the solution set. That is, tell whether there exists a unique solution no
solution or infinitely many solutions.
- A is a 5 × 6 matrix, rank = 4 and rank= 4 . Hint: This says b is in the span of four of the columns. Thus the columns are not independent.
- A is a 3 × 4 matrix, rank = 3 and rank= 2 .
- A is a 4 × 2 matrix, rank = 4 and rank= 4 . Hint: This says b is in the span of the columns and the columns must be independent.
- A is a 5×5 matrix, rank = 4 and rank= 5 . Hint: This says b is not in the span of the columns.
- A is a 4 × 2 matrix, rank = 2 and rank= 2.

- A is a 5 × 6 matrix, rank
- Suppose A is an m × n matrix in which m ≤ n. Suppose also that the rank of A equals m.
Show that A maps F
^{n}onto F^{m}. Hint: The vectors e_{1},,e_{m}occur as columns in the row reduced echelon form for A. - Suppose A is an m × n matrix in which m ≥ n. Suppose also that the rank of A equals n. Show that A is one to one. Hint: If not, there exists a vector, x such that Ax = 0, and this implies at least one column of A is a linear combination of the others. Show this would require the column rank to be less than n.
- Explain why an n × n matrix A is both one to one and onto if and only if its rank is n.
- If you have not done this already, here it is again. It is a very important result. Suppose A is
an m × n matrix and B is an n × p matrix. Show that
Hint: Consider the subspace, B

∩ kerand suppose a basis for this subspace is. Now supposeis a basis for ker. Letbe such that Bz_{i}= w_{i}and argue thatHere is how you do this. Suppose ABx = 0. Then Bx ∈ ker

∩ Band so Bx = ∑_{i=1}^{k}Bz_{i}showing that - Recall that every positive integer can be factored into a product of primes in a unique way. Show there must be infinitely many primes. Hint: Show that if you have any finite set of primes and you multiply them and then add 1, the result cannot be divisible by any of the primes in your finite set. This idea in the hint is due to Euclid who lived about 300 B.C.
- There are lots of fields. This will give an example of a finite field. Let ℤ denote the set of
integers. Thus ℤ = . Also let p be a prime number. We will say that two integers, a,b are equivalent and write a ∼ b if a−b is divisible by p. Thus they are equivalent if a − b = px for some integer x. First show that a ∼ a. Next show that if a ∼ b then b ∼ a. Finally show that if a ∼ b and b ∼ c then a ∼ c. For a an integer, denote bythe set of all integers which is equivalent to a, the equivalence class of a. Show first that is suffices to consider onlyfor a = 0,1,2,,p − 1 and that for 0 ≤ a < b ≤ p − 1,≠. That is,=where r ∈. Thus there are exactly p of these equivalence classes. Hint: Recall the Euclidean algorithm. For a > 0, a = mp + r where r < p. Next define the following operations.=and=, then+=+with a similar conclusion holding for multiplication. Thus for addition you need to verify=and for multiplication you need to verify=. For example, if p = 5 you have=and=. Is=? Is=? Clearly so in this example because when you subtract, the result is divisible by 5. So why is this so in general? Now verify thatwith these operations is a Field. This is called the integers modulo a prime and is written ℤ
_{p}. Since there are infinitely many primes p, it follows there are infinitely many of these finite fields. Hint: Most of the axioms are easy once you have shown the operations are well defined. The only two which are tricky are the ones which give the existence of the additive inverse and the multiplicative inverse. Of these, the first is not hard. −=. Since p is prime, there exist integers x,y such that 1 = px + ky and so 1 − ky = px which says 1 ∼ ky and so=. Now you finish the argument. What is the multiplicative identity in this collection of equivalence classes? Of course you could now consider field extensions based on these fields. - Suppose the field of scalars is ℤ
_{2}described above. Show thatThus the identity is a comutator. Compare this with Problem 50 on Page 534.

- Suppose V is a vector space with field of scalars F. Let T ∈ℒ, the space of linear transformations mapping V onto W where W is another vector space. Define an equivalence relation on V as follows. v ∼ w means v − w ∈ ker. Recall that ker≡. Show this is an equivalence relation. Now foran equivalence class define T
^{′}≡ Tv. Show this is well defined. Also show that with the operationsis a vector space. Show next that T^{′}: V∕ker→ W is one to one, linear, and onto. This new vector space, V∕keris called a quotient space. Show its dimension equals the difference between the dimension of V and the dimension of ker. - Let V be an n dimensional vector space and let W be a subspace. Generalize the above problem to define and give properties of V∕W. What is its dimension? What is a basis?
- If F and G are two fields and F ⊆ G, can you consider G as a vector space with field of scalars F? Explain.
- Let A denote the real roots of polynomials in ℚ. Show A can be considered a vector space with field of scalars ℚ. What is the dimension of this vector space, finite or infinite?
- As mentioned, for distinct algebraic numbers α
_{i}, the complex numbers_{i=1}^{n}are linearly independent over the field of scalars A where A denotes the algebraic numbers, those which are roots of a polynomial having integer (rational) coefficients. What is the dimension of the vector space ℂ with field of scalars A, finite or infinite? If the field of scalars were ℂ instead of A, would this change? What if the field of scalars were ℝ? - Suppose F is a countable field and let A be the algebraic numbers, those numbers in G which
are roots of a polynomial in F. Show A is also countable.
- This problem is on partial fractions. Suppose you have
where the polynomials q

_{i}are relatively prime and all the polynomials pand q_{i}have coefficients in a field of scalars F. Thus there exist polynomials a_{i}having coefficients in F such thatExplain why

Now continue doing this on each term in the above sum till finally you obtain an expression of the form

Using the Euclidean algorithm for polynomials, explain why the above is of the form

where the degree of each r

_{i}is less than the degree of q_{i}and Mis a polynomial. Now argue that M= 0 . From this explain why the usual partial fractions expansion of calculus must be true. You can use the fact that every polynomial having real coefficients factors into a product of irreducible quadratic polynomials and linear polynomials having real coefficients. This follows from the fundamental theorem of algebra in the appendix. - Suppose is an independent set of smooth functions defined on some interval. Now let A be an invertible n × n matrix. Define new functionsas follows.
Is it the case that

is also independent? Explain why. - A number is transcendental if it is not the root of any nonzero polynomial with rational
coefficients. As mentioned, there are many known transcendental numbers. Suppose α is a
real transcendental number. Show that is a linearly independent set of real numbers if the field of scalars is the rational numbers.

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