The n times n matrices of determinant zero The n times n invertible matrices If
ID: 3008320 • Letter: T
Question
The n times n matrices of determinant zero The n times n invertible matrices If A is an m times n matrix and B is a nonzero element of R^m, do the solutions to the system AX = B form a subspace of R^n? Why or why not? Complex numbers a + bi where a and b are integers are called Gaussian integers. Do the Gaussian integers form a subspace of the vector space of complex numbers? Why or why not? Do the sequences that converge to zero form a subspace of the vector space of convergent sequences? How about the sequences that converge to a rational number? Do the series that converge to a positive number form a subspace of the vector space of convergent series? How about the series that converge absolutely?Explanation / Answer
5) A is a mxn matrix and B is a non zero matrix in R^m
If m = n and A is not singular, then a unique solution exists
If m < n, then dependent solutions exist which form a vector space.
If m >n the system have more equations than variables and may not be consistent always
Hence in general, cannot form a subspace.
6) Complex numbers of the form a+bi
where a and b are integers
Consider a+bi and c+di
a+c+i(b+d) is the sum where a+c and b+d are also integers
Hence addition is closed
Addition is also associative since integers addition is associative
Identity is 0+0.i for all a+bi
Inverse of a+bi is -a-bi in V since -a and -b are integers
Multiplication:
(a+bi)(c+di) = ac-bd+i(bc+ad)
ac-bd is an integer and also bc+ad So multiplication is closed
c(a+bi) = ca+cbi again in V
Hence these a+bi's where a and b are Gaussian integers form a vector space.