Basics for Math 18D, borrowed from earlier class
... (9) Nul (T ) = {v ∈ V : T (v) = 0} – all vectors in the domain which are sent to zero. (a) T is one to one iff Nul (T ) = {0} . (b) be able to find a basis for Nul (A) ⊂ Rn when A is a m × n matrix. (10) Ran (T ) = {T (v) ∈ W : v ∈ V } – the range of T. Equivalently, w ∈ Ran (T ) iff there exists a ...
... (9) Nul (T ) = {v ∈ V : T (v) = 0} – all vectors in the domain which are sent to zero. (a) T is one to one iff Nul (T ) = {0} . (b) be able to find a basis for Nul (A) ⊂ Rn when A is a m × n matrix. (10) Ran (T ) = {T (v) ∈ W : v ∈ V } – the range of T. Equivalently, w ∈ Ran (T ) iff there exists a ...
MATHEMATICAL METHODS SOLUTION OF LINEAR SYSTEMS I
... The rank of a matrix is always unique. The rank of a zero matrix is always zero. The rank of a non –singular matrix of order “n” is equals to “n”. The rank of a singular matrix of order “n” is less than “n”. The rank of a unit matrix of order “n” is equals to “n”. If A is a matrix of ord ...
... The rank of a matrix is always unique. The rank of a zero matrix is always zero. The rank of a non –singular matrix of order “n” is equals to “n”. The rank of a singular matrix of order “n” is less than “n”. The rank of a unit matrix of order “n” is equals to “n”. If A is a matrix of ord ...
Linear algebra and the geometry of quadratic equations Similarity
... Similarity transformations and orthogonal matrices First, some things to recall from linear algebra. Two square matrices A and B are similar if there is an invertible matrix S such that A = S −1 BS. This is equivalent to B = SAS −1 . The expression SAS −1 is called a similarity transformation of the ...
... Similarity transformations and orthogonal matrices First, some things to recall from linear algebra. Two square matrices A and B are similar if there is an invertible matrix S such that A = S −1 BS. This is equivalent to B = SAS −1 . The expression SAS −1 is called a similarity transformation of the ...
![[Review published in SIAM Review, Vol. 56, Issue 1, pp. 189–191.]](http://s1.studyres.com/store/data/017041202_1-65d6c90f40ee4c0571100dd75b61a7f5-300x300.png)
math21b.review1.spring01
... of its dimension as mxn where m is the number of rows (number of entries in each column) and n is the number of columns (number of entries in each row); a vector is simply an mx1 matrix Rn – is the space defined by ordered “n-tuples” of real numbers (e.g. (x1, x2, …, xn) ) Linear transformation – a ...
... of its dimension as mxn where m is the number of rows (number of entries in each column) and n is the number of columns (number of entries in each row); a vector is simply an mx1 matrix Rn – is the space defined by ordered “n-tuples” of real numbers (e.g. (x1, x2, …, xn) ) Linear transformation – a ...
