LA

If an m x n matrix A has a pivot position in every row, then the equation Ax = b has a unique solution for each b in Rm. (T/F)

False. If A has m pivots and n variables with n > m then there is at least one free variable, and an infinite number of solutions for Ax = b

Automatic Linear Dependence (m x n)

If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set {v1, ..., vp} in Rn is linearly dependent if p > n.

An outcome from Row Equivalence

If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set.

Parallelogram Rule for Addition

If u and v in R2 are represented as points in the plane, then u + v corresponds to the fourth vertex of the parallelogram whose other vertices are u, 0, and v.

The fasted way to confirm if T is not linear...?

Is T(0) = 0. If not, then T is not linear. If it is, it tells you nothing so you then need to apply (i) and (ii)

Theorem 12 Supporting One-to-One:

Let T : Rn -> Rm be a linear transformation, and let A be the standard matrix for T. Then: T is one-to-one if and only if the columns of A are linearly independent. By Theorem 11 T is one-to-one if and only if Ax = 0 had only the trivial solution. This happens if and only if the columns are linearly independent. A pivot in every column

Theorem 12 Supporting onto

Let T : Rn -> Rm be a linear transformation, and let A be the standard matrix for T. Then: T maps Rn onto Rm if and only if the columns of A span Rm. By Theorem 4 in Section 1.4, the columns of A span Rm if and only if for each b in Rm the equation Ax = b is consistent—in other words, if and only if for every b, the equation T(x) = b has at least one solution. This is true if and only if T maps Rn onto Rm.

Theorem 11 Supporting One-to-One:

Let T : Rn -> Rm be a linear transformation. Then T is one-to-one if and only if the equation T(x) = 0 has only the trivial solution.

Discuss vector and span v, then u and v in R3

Let v be a nonzero vector in R3. Then Span {v} is the set of all scalar multiples of v, which is the set of points on the line in R3 through v and 0. If u and v are nonzero vectors in R3, with v not a multiple of u, then Span {u; v} is the plane in R3 that contains u, v, and 0. In particular, Span {u; v} contains the line in R3 through u and 0 and the line through v and 0.

Let T be a linear transformation from Rn to Rm. Moreover let T be both one to one and onto. Describe the relationship between n and m

n = m

What is a pivot position / column

A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A. A pivot column is a column of A that contains a pivot position.

Linear Independence for a Set containing only one vector, say v.

A set containing only one vector—say, v—is linearly independent if and only if v is not the zero vector. This is because the vector equation x1v = 0 has only the trivial solution when v <> 0. The zero vector is linearly dependent because x10 = 0 has many nontrivial solutions.

Linear Independence for a Set containing two vectors {v1, v2}

A set of two vectors {v1, v2} is linearly dependent if at least one of the vectors is a multiple of the other. The set is linearly independent if and only if neither of the vectors is a multiple of the other. In geometric terms, two vectors are linearly dependent if and only if they lie on the same line through the origin

What is a solution

A solution of the system is a list (s_1, s_2, ..., s_n) of numbers that makes each equation a true statement when the values are substituted i.e., s_n substituted for x_n

Name the 3 Elementary Row Operations

1. (Replacement) Replace one row by the sum of itself and a multiple of another row. 2. (Interchange) Interchange two rows. 3. (Scaling) Multiply all entries in a row by a nonzero constant.

If A is an m x n matrix, u and v are vectors in Rn, and c is a scalar, then...?

1. A(u + v) = Au + Av 2. A(cu) = c(Au)

Name the 3 properties of a matrix in echelon form

1. All nonzero rows are above any rows of all zeros. 2. Each leading entry of a row is in a column to the right of the leading entry of the row above it. 3. All entries in a column below a leading entry are zeros.

Theorem 4: Let A be an m x n matrix. recall the following statements that are logically equivalent. That is, for a particular A, either they are all true statements or they are all false.

1. For each b in Rm, the equation Ax = b has a solution. 2. Each b in Rm is a linear combination of the columns of A. 3. The columns of A span Rm. 4. A has a pivot position in every row.

Name the 3 characteristics of a system of linear equations

1. No solution exists, or 2. Exactly one solution exists, or 3. Infinitely many solutions exist

Define the steps required to WRITE THE SOLUTION SET (OF A CONSISTENT SYSTEM) IN PARAMETRIC VECTOR FORM

1. Row reduce the augmented matrix to reduced echelon form. 2. Express each basic variable in terms of any free variables appearing in an equation. 3. Write a typical solution x as a vector whose entries depend on the free variables, if any. 4. Decompose x into a linear combination of vectors (with numeric entries) using the free variables as parameters.

Name the 2 additional properties of reduced echelon form

1. The leading entry in each nonzero row is 1. 2. Each leading 1 is the only nonzero entry in its column.

Define the steps for USING ROW REDUCTION TO SOLVE A LINEAR SYSTEM

1. Write the augmented matrix of the system. 2. Use the row reduction algorithm to obtain an equivalent augmented matrix in echelon form. Decide whether the system is consistent. If there is no solution, stop; otherwise, go to the next step. 3. Continue row reduction to obtain the reduced echelon form. 4. Write the system of equations corresponding to the matrix obtained in step 3. 5. Rewrite each nonzero equation from step 4 so that its one basic variable is expressed in terms of any free variables appearing in the equation.

What is a nonzero row?

A nonzero row or column in a matrix means a row or column that contains at least one nonzero entry

What is a vector in R3

A 3 x 1 column matrix with three entries. They are represented geometrically by points in a three-dimensional coordinate space

What is a system of linear equations

A collection of one or more linear equations involving the same variables

What is a leading entry?

A leading entry of a row refers to the leftmost nonzero entry (in a nonzero row)

What is the Existence and Uniqueness Theorem

A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column—that is, if and only if an echelon form of the augmented matrix has no row of the form [0, 0, ..., b] If a linear system is consistent, then the solution set contains either (i) a unique solution, when there are no free variables, or (ii) infinitely many solutions, when there is at least one free variable.

There is a linear transformation T: R → R that sends the interval [−1, 1] to the interval [1, 3]

A linear transformation T : R → R always has the form T(x) = ax, for some scalar a. It follows that T takes the interval [−1, 1] to the interval [−a, a].

Discuss a one-to-one mapping

A mapping T : Rn -> Rm is said to be one-to-one if each b in Rm is the image of at most one x in Rn. Equivalently, T is one-to-one if, for each b in Rm, the equation T(x) = 0 has either a unique solution or none at all. "Is T one-to-one?" is a uniqueness question. The mapping T is not one-to-one when some b in Rm is the image of more than one vector in Rn. If there is no such b, then T is one-to-one. Big to small (R5 to R3) never 1:1 pivot in every column = 1:1

Discuss an onto mapping

A mapping T : Rn -> Rm is said to be onto Rm if each b in Rm is the image of at least one x in Rn. Equivalently, T is onto Rm when the range of T is all of the codomain Rm. That is, T maps Rn onto Rm if, for each b in the codomain Rm, there exists at least one solution of T(x) = b. "Does T map Rn onto Rm?" is an existence question. The mapping T is not onto when there is some b in Rm for which the equation T(x) = b has no solution

If u, v and w are nonzero vectors in R2, then w must be a linear combination of u and v. (T/F)

False. All three vectors may be linearly independent.

If A is an m x n matrix and the equation Ax = b is consistent for some b, then the columns of A span Rm. (T/F)

False. Ax = b must be consistent for ALL b

What is an Homogeneous Linear Systems

A system of linear equations is said to be homogeneous if it can be written in the form Ax = 0, where A is an m x n matrix and 0 is the zero vector in Rm. Such a system Ax = 0 always has at least one solution, namely, x = 0 (the zero vector in Rn). This zero solution is usually called the trivial solution

Define a Transformation T

A transformation (or function or mapping) T from Rn to Rm is a rule that assigns to each vector x in Rn a vector T(x) in Rm. The set Rn is called the domain of T, and Rm is called the codomain of T. The notation T : Rn -> Rm indicates that the domain of T is Rn and the codomain is Rm. For x in Rn, the vector T(x) in Rm is called the image of x (under the action of T). The set of all images T(x) is called the range of T. The range of T is the set of all linear combinations of the columns of A, because each image T(x) is of the form Ax.

What is the most important class of transformations in linear algebra?

A transformation (or mapping) T is linear if: (i) T(u + v) = T(u) + T(v) for all u, v in the domain of T; (ii) T(cu) = cT(u) for all scalars c and all u in the domain of T.

What is a vector?

A vector is an ordered list of numbers, A matrix with only one column is called a column vector, or simply a vector

Always, Sometimes, Never: If a is constant the T(x) = ax is a linear transformation from R to R

Always. This is a scaling operation, the dimensions will not change from R.

What is an augmented matrix

An augmented matrix of a system consists of the coefficient matrix with an added column containing the constants from the right sides of the equations.

Characterization of Linearly Dependent Sets

An indexed set S = {v1, ..., vp} of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others. In fact, if S is linearly dependent and v1 <> 0, then some vj (with j > 1) is a linear combination of the preceding vectors, v1, ..., vj-1.

Comment on the Uniqueness of the Reduced Echelon Form

Each matrix is row equivalent to one and only one reduced echelon matrix.

If none of the vectors in the set S = {v1; v2; v3} in R3 is a multiple of one of the other vectors, then S is linearly independent. (T/F)

False. Consider if the third vector is the sum of the first two...

Let A and B be any two n x n matrices. Let "rank" be the number of pivots of a matrix. Then rank(A+B) = rank(A) + rank(B). (T/F)

False. Despite not knowing the detail of Rank its reasonable to assume the pivots can change in matrix addition.

The equation Ax = 0 has the trivial solution if and only if there are no free variables. (T/F)

False. Every equation Ax = 0 has the trivial solution.

True / False: Any function T with the property T(cx) = cT(x) for all scalars c and vectors x is a linear combination

False. It must satisfy both conditions. The one stated and T(u) + T(v) = T(u + v)

If S and T are functions from R 2 → R 2 that are both nonlinear, then S ◦ T must also be nonlinear. (T/F)

False. There is a circumstance where this may produce the identity matrix, which is linear.

True / False: If A is a 3 x 2 matrix then the columns of A are linearly independent.

False. While this is possible, it will not be in every occurrence. For a 3 x 2 matrix to be linearly independent it must have a pivot in every column. This is possible but not in all cases.

If vector u is a linear combination of vectors v and w, then w must be a linear combination of u an v. (T/F)

False. u and v may be linearly independent.

What is a Linear Combination?

Given vectors v1, v2, ..., vp in Rn and given scalars c1, c2, ..., cp, the vector y defined by is called a linear combination

Automatic Linear Dependence (the zero vector)

If a set S = {v1, ..., vp} in Rn contains the zero vector, then the set is linearly dependent.

Can a set of two vectors span R3?

No. a 3 x 2 matrix can only have at most 2 pivots. Since there are three rows at least one does not contain a pivot so the columns do not span R3. Theorem 4.

Always, Sometimes, Never: A consistent system of three linear equations in two variables has a free variable.

Sometimes true. Assuming a 3x3 augmented matrix of the form x_1, x_2 and the constant solution, with 2 coefficients the second variable may be free, but not always. As there is no further detail on the echelon form we can conclude that in reduced echelon form we may have 1 pivot column and a free variable, or 2 pivot columns. 3 Pivot columns would define an inconsistent system.

Always, Sometimes, Never: If a set S{} of vectors in Rn is linearly dependent then S{} contains more than n vectors.

Sometimes. The set of S vectors in Rn may be linearly dependent as a result of this condition, however this is not the ONLY reason that a set is linearly dependent. For example S may contain less than n vectors, where one vector is the zero vector such that {v,0}. As a result S is determined as linearly dependent for other unrelated reasons.

Comment on the geographic translation on nonhomogeneous systems

Suppose the equation Ax = b is consistent for some given b, and let p be a solution. Then the solution set of Ax = b is the set of all vectors of the form w = p + v_h, where v_h is any solution of the homogeneous equation Ax = 0.

Sum it up with a matrix in R3 :)

The columns of a 3 × 3 matrix A span R 3 if and only if there is a pivot in every row of A. The columns are linearly independent if and only if there is a pivot in every column.

Linear Independence of Matrix Columns (Key Concept)

The columns of a matrix A are linearly independent if and only if the equation Ax = 0 has only the trivial solution.

The definition of Ax leads directly to what useful fact?

The equation Ax = b has a solution if and only if b is a linear combination of the columns of A.

When does an homogenous linear system have a nontrivial solution

The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable.

What is a solution set, and how does it relate to two systems being equivalent?

The set of all possible solutions is called the solution set of the linear system. Two linear systems are called equivalent if they have the same solution set.

If S and T are both linear transformations from R 2 to R 2 , then S ◦ T is also a linear transformation from R 2 to R 2. (T/F)

This is True. Given the linearity this passes both tests (i) and (ii)

Two fundamental questions about a linear system involve existence and uniqueness (T/F)

True

If an n x n matrix A has n pivot positions, then the reduced echelon form of A is the n x n identity matrix. (T/F)

True. Based on the definition of an identity matrix and RREF

If A and B are two n x n matrices with n pivots, then A can be transformed into B by means of elementary row operations. (T/F)

True. By the conditions of Row Equivalence and unique nature of row reduced echelon form.

True / False: If v is a nonzero vector in Rn then span{v, -3v} is a line.

True. The defined span is comprised of v and -3 multiplied by v. In this case it can be considered that -3v is in the span of v, 3 times longer than v and pointing in the opposite direction.

If v and w are vectors in R4, then the zero vector in R4 must be a linear combination of v and w. (T/F)

True. The zero vector is linearly dependent.

If v and w are vectors in R4 then v must be a linear combination of v and w. (T/F)

True. Tricky one - consider (1).v + (0)w would make v a linear combination.

If A is a 4x4 matrix and the system Ax = b has only the unique solution, then the system Ax = 0 has only the solution x = 0. (T/F)

True. Watch this one. The question can replace b with an actual vector to confuse, but the answer reverts to the standard criteria of the unique solution (the trivial solution) where x = 0. The compliment is also true, If a system Ax = b has more than one solution, then so does the system Ax = 0.

Define Row Equivalence

Two matrices are called row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other. It is important to note that row operations are reversible. Since they are row equivalent, they have the same reduced echelon form

Define the shape of a matrix

an m x n matrix is a rectangular array of numbers with m rows and n columns. (The number of rows always comes first.)