Linear Algebra Midterm #1

coordinate

scalars in a vector (4th coordinate = 4th number in vector)

a matrix is…

m x n (rows x columns)

(i,j)

(row, column) method to indicate a specific scalar in a matrix

transpose

interchange the rows and columns of a matrix where the (i,j) notation is also flipped for each entry

linear operation

the operation can be distributed normally (transposition, trace)

involution

Do it once, it changes; do it twice, it goes back to the original

(A^T)^T= A

Examples: transposition

symmetric matrix

S^T=S

number of rows and columns are equal

1st row = 1st col, 2nd row = 2nd col, etc

symmetric across the diagona

trace

sum of a matrix’s diagonal

trace under transposition…

stays the same (stable)

diagonal

collection of (i,i) entries

upper triangular

every entry below the diagonal is zero (a_ij=0 for i>j)

lower triangular

every entry above the diagonal is zero (a_ij=0 for i<j)

diagonal matrix

every non diagonal entry is zero

diag(d1,…,dn)

is the nxn diagonal matrix with diagonal d1,…,dn (blank entries understood to be zero)

identity matrix

has ones on the diagonal and zeros elsewhere

standard basis…

{e1,…,en} of Rⁿ are the columns of the identity matrix (I_n)

each e1, e2, etc is a column of the identity

pivot column

the first nonzero column

a matrix is rank one if…

every column is a multiple of the first pivot column