4.2 Flashcards
T or F: Let A denote an m x n matrix.
A null space is a vector space
The statement is true. The null space of an m x n matrix A is a subspace of set of real numbers R^n.
T or F: Let A denote an m x n matrix.
The column space of an m x n matrix is in set of real numbers R^m.
The statement is true. The column space of an m x n matrix A is a subspace of set of real numbers R^m.
T or F: Let A denote an m x n matrix.
The column space of A, Col A, is the set of all solutions of Ax = b.
The statement is false. The column space of A is Col A = {b : b = Ax for some x in set of real numbers R^n}.
T or F: Let A denote an m x n matrix.
The null space of A, Nul A, is the kernel of the mapping x maps to Ax.
The statement is true. The kernel of a linear transformation T, from a vector space V to a vector space W, is the set of all u in V such that T(u) = 0. Thus, the kernel of a matrix transformation T(x) = Ax is the null space of A.
T or F: The range of a linear transformation is a vector space.
The statement is true. The range of a linear transformation T, from a vector space V to a vector space W, is a subspace of W.
T or F: The set of all solutions of a homogeneous linear differential equation is the kernel of a linear transformation.
The statement is true. The linear transformation that corresponds to a linear differential equation maps each function f to a linear combination of f and at least one of its derivatives. For f to satisfy a homogeneous differential equation, this transformation must map f to 0. Thus, the set of all solutions is the kernel of the corresponding linear transformation.
T or F: The row space of A Superscript T is the same as the column space of A.
The statement is true because the rows of A^T are the columns of (A^T)^T = A.
Definition for Null Space
Nul A = {x: x is in R^n and Ax = 0}
Thrm: The null space of m x n matrix A is a ________ of R^n. Equivalently, the set of all solutions to a system Ax = 0 of m homogeneous linear eqns in n unknowns is a ________ of R^n.
subspace
Definition of Column Space
Col A = Span {a_1,……..,a_n}
or
Col A = {b : b = Ax for some x in R^n}
The Column Space of an m x n matrix A is a ________ of R^m.
subspace
Null Space or Col Space?
Is a subspace of R^n.
Null Space
Null Space or Col Space?
Is a subspace of R^m
Col Space
Null Space or Col Space?
Implicity defined; that is, you are given only a condition (Ax=0) that vectors in ____ must satisfy
Null Space
Null Space or Col Space?
Explicited defined; that is, you are told how to build vectors in ______
Col Space
Null Space or Col Space?
It takes time to find vectors in this subspace. Row operations on [A 0] are required.
Null Space
Null Space or Col Space?
Easy to find vectors in this subspace. The columns of A are displayed; other are formed from them
Col Space
Null Space or Col Space?
There is no obvious relation b/w this subpsace and the entries in A
Null Space
Null Space or Col Space?
There is an obvious relation b/w this subspace and the entries in A, since each column of A is in Col A.
Col Space
Null Space or Col Space?
A typical vector v in this subspace has the property that Av = 0
Null Space
Null Space or Col Space?
A typical vector v in this subspace has the propety that the eqn Ax = v is consistent
Col Space
Null Space or Col Space?
Given a specific vector v, it is easy to tell if v is in this subspace. Just Compute Av
Null Space
Null Space or Col Space?
Given a specific vector, it may take time to tell if v is in this subspace. Row operations on [A v] are required
Col space
Null Space or Col Space?
This subspace = {0} if and only if the eqn Ax = 0 has only the trivial solution
Null space
Null Space or Col Space?
This subspace = R^m if and only if the eqn Ax = b has a soln for every b in R^m
Col Space
Null Space or Col Space?
This subspace = {0} if and only if if the linear transformation x maps to Ax is one to one
Null space
Null Space or Col Space?
This subspace = R^m if and only if the linear transformation x maps to Ax maps R^n onto R^m
Col Space
Definition of a linear transformation from two vector spaces
A linear transformation T from a vector space V into a vector space W is a rule that assigns to each vector x in V a unique vector T(x) in W, such that:
i. T(u+v) = T(u) + T(v) for all u, v in V, and
ii, T(cu) = cT(u) for all u in V and all scalars c.
what is the kernel?
The kernel of a linear transformation is the set of x in V. The set of x in V such that T(x) = 0.
