The Basics (all sections)
For any linear system of simultaneous equations, these are the only three possible outcomes for the solution set.
WHAT ARE:
Inconsistent System: No solution
Consistent System: Unique solution or infinitely many solutions
For a system represented by the matrix equation Ax = b, if the system is consistent then we can say this about b relative to the matrix A.
WHAT IS: The vector b is within the span of the columns of A.
This test is very useful for proving that a set of vectors, or columns of a matrix, are linearly dependent, but can not, in general, be used to prove independence.
What is: "The pairwise scalar multiple test" (my name for it not the book's) - i.e. the unnamed/unnumbered green textbox on page 62.
Rationale: This test only works for a PAIR of vectors, i.e. a set of 2. For a set with p > 2 vectors, or a matrix with n > 2 columns, you can "pass" this test and still have a linearly dependent set.
Alternate Answer: Theorem 8. It can only be used to prove dependence (when p > n, or equivalently, when n > m for a matrix). THE CONVERSE OF THEOREM 8 IS NOT TRUE IN GENERAL.
This property is both necessary and sufficient for a linear transformation T(x) = Ax to map ONTO Rm
WHAT IS: The columns of A span Rm per Theorem 12a
ALTERNATE ANSWER:
"An echelon form of A (or its REF) must have a pivot in every row" which proves that the columns of A span Rm based on Theorem 4 (d) <==> (c)
TRUE or FALSE:
If the system Ax=b has no free variables, there will be a unique solution for x.
What is FALSE?
If the system is inconsistent there will be no solution.
TRUE OR FALSE: We have to be careful when using elementary row operations (EROs) that we don't change the solution set of the system.
WHAT IS FALSE (but it's also sorta true)
PROPERLY executed EROs do NOT change the solution set, the resulting new system has exactly the same solution set as the original.
However, it IS true that we do need to BE VERY CAREFUL that we execute the EROs properly (e.g. no algebra or arithmetic errors) or we almost certainly will change the solution set.
For a consistent system represented by the matrix equation Ax = b, with A being an mxn matrix, this is the geometric interpretation of the solution set when there are
(1) no free variables;
(2) a single free variable;
(3) two free variables
WHAT IS:
1) a point in Rm, i.e. a unique solution (a zero-dimensional solution)
2) a line in Rm (a one-dimensional solution)
3) a plane in Rm (a two-dimensional solution)
NOTE: Even though the solutions are 0, 1, and 2-dimensional respectively, they may reside in a higher dimensional vector space
This simple test can be used to prove that a set of vectors, or columns of a matrix, are linearly dependent.
What is: "Does the set contain the zero vector"
ALTERNATE ANSWERS:
1) What is the pairwise scalar multiple test.
2) If you happen to observe a linear dependence relation between the vectors, although sometimes that can be difficult to spot
This property is both necessary and sufficient for a linear transformation T(x) = Ax to map 1-to-1 from Rn to Rm
WHAT IS: The columns of A must be linearly independent per Theorem 12b
ALTERNATE ANSWERS:
1) The equation T(x) = Ax = 0 has only the trivial solution per Theorem 11
2) An echelon form of A has a pivot in every column so there are no free variables
3) If the system defined by the equation T(x) = Ax = b is consistent, then there will be a unique solution.
Parametric solutions for the vector x for the system Ax=b are expressed in terms of these type of variables.
What are FREE variables?
It is OK to leave off the square brackets of a matrix and just have a sea of numbers scattered all over the page.
WHAT IS HELL NO!!
No brackets, no matrix, no credit, no brainer.
TRUE OR FALSE: Given matrix equation Ax = b, if an echelon form of A has at least one column without a pivot, i.e. there is a free variable(s), then there are an infinite number of solutions.
WHAT IS FALSE!!!!
Rationale: MOST but not all of the time this is that is the case (infinite number of solutions). However, the system may be inconsistent and therefore have NO solutions.
TRUE OR FALSE: If an echelon form of an mxn matrix A contains a row of all zeros then the columns are linearly dependent.
WHAT IS FALSE!!!!
A ROW of all zeros is NOT the zero vector. A row of all zeros does NOT preclude having a pivot in every column. For example:
[[1,0],[0,1],[0,0]]
A linear transformation T that maps Rn to Rm with n>m (more columns than rows) can NOT have this property.
WHAT IS one-to-one?
Rationale: T maps from a higher dimensional vector space (Rn) to a lower dimensional one (Rm). Rm has infinitely more vectors that Rn so it CAN "fill the bucket" (but may not unless T is ONTO Rm) but will have an infinite number of vectors left over and thus will have to assign multiple x vectors to a single b vector.
(The "bucket" reference is to the diagram in the supplement posted on Blackboard)
For a linear transformation T(x)=Ax, if T(x)=b we say that b is this relative to x.
What is that b is the image of x under the transformation T?
Including the vertical bar in an augmented matrix denoting where the equal sign is, i.e. separating the coefficients of the variables from the constants in the b vector on the right-hand side (RHS) of the equation, is mandatory.
WHAT IS NO?
I HIGHLY recommend that you include the vertical bar, but it is by no means mandatory. To wit, the book does not use that notation, although I wish they would, or would at least mention it.
For a consistent system represented by the matrix equation Ax = b, when there are free variable(s), we need to use these two technique to find the solution set for x.
WHAT IS:
1) Row reduce the matrix A to echelon form or REF
2) Use Back Substitution to solve for x in terms of its free variable(s)
NO BASIC VARIABLES SHOULD EVER APPEAR IN A PARAMETRIC SOLUTION
TRUE or FALSE: An mxn matrix can either be linearly independent or dependent.
WHAT IS FALSE!!!
Rationale: Linear independence/dependence is fundamentally defined for a set of vectors (Chapter 1.7 equation (2)) NOT a matrix. The COLUMNS of a matrix consist of a set of vectors that can have this property, but the matrix A itself does not.
A linear transformation T that maps Rn to Rm with m>n (more rows than columns) can NOT have this property.
WHAT IS "Onto"?
Rationale: T maps from a lower dimensional vector space (Rn) to a higher dimensional one (Rm). Rm has infinitely more vectors that Rn so it is impossible to "fill the bucket" - there just are not enough vectors in Rn to do so. It is POSSIBLE that the mapping is 1-to-1, but it is NOT guaranteed.
(The "bucket" reference is to the diagram in the supplement posted on Blackboard)
TRUE of FALSE:
If no columns of a matrix are a scalar multiple of another, then the columns form a linearly independent set.
What is FALSE!
The "pairwise scalar multiple test" can only prove independence for a set of two vectors. I can, however, prove dependence for a set of any number of vectors.
TRUE OR FALSE:
There is no need to document your row operations. They are self-explanatory and any idiot, including our instructor, should be able to figure out what I did. As long as I know what I am doing/did, it doesn't matter if anyone else does.
WHAT IS TRUE if you: (1) want to maximize your chance of making an error; (2) make it harder for you to figure out what you did wrong if you have to error check your work; (3) you don't want full credit or to maximize your partial credit
WHAT IS FALSE if you are smart.
I actually do look at everything you write in hopes that I can find a good solid solution, and nothing that is egregiously wrong. When things do go wrong, I try hard to figure out where/why to: (1) properly determine partial credit; (2) give you helpful feedback.
If you want to (1) make it hard for me to give you credit for what you know; (2) allow a small error to appear like a big one because I can't figure out what the hell you were trying to do and why; (3) don't need any feedback to identify and help resolve issues ==> then be my guest and don't bother documenting your EROs. :)
TRUE or FALSE: All basic variables are a function of one or more free variables.
WHAT IS FALSE:
It is possible to have a basic variable be independent of the free variable(s). For example: x3 is the free variable in the system below and the basic variable x4 = 2 completely independent of x3.
[[1,0,1,0,|,2],[0,1,2,3,|,4],[0,0,0,1,|,2]]
One way to interpret the algebraic definition for linear independence of a set of vectors in a quasi-geometric context is that you can't ____
WHAT IS that you can't build any of the vectors in the set using a linear combination of any or all of the others.
Suppose you have p vectors and you want to try to build vp from the others. You would solve the equation given in the definition of linear independence for the vector vp.
In doing so, we will divide the equation by xp and we can safely assume that it does not equal 0 because otherwise it can be removed from the original equation from the definition.
If the set of vectors is independent, there will be no set of xi coefficients that will satisfy our attempt to "build vp" as follows:
bbv_p = (x_1bbv_1+... + x_(p-1) bbv_(p-1))/x_p
TRUE OR FALSE: For a linear transformation T(x) = Ax where A is a square (n x n) matrix, it is possible for the mapping to be ONTO Rn but not 1-to-1
WHAT IS FALSE!!!!
Rationale: A square matrix has an equal number of columns and rows. Having a pivot in every column (or not) is equivalent to having a pivot in every row (or not). Therefore, the conditions of Theorem 12 (a) and (b) are either BOTH satisfied or NEITHER are satisfied.
This is how many points you will score on your exam if you exceed the duration time limit or if it submitted after the submission deadline.
WHAT IS ZERO, ZILCH, NADA, SQUAT?
That's right, I have been fairly understanding of the various issues that can come up in taking a free response exam (quiz) online. I KNOW stuff happens and for some students this is a relatively new experience. However, by now, everyone should have figured out how to manage their time and deal with the logistics of submitting their file(s), properly named and formatted, ON TIME. There will be no grace period or exceptions unless there are EXTRAORDINARY CIRCUMSTANCES (such as the DMV is hit with a large meteor or something of that magnitude).