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# [M] Find a column of the matrix in Exercise 40 that can be

ISBN: 9780321385178 62

## Solution for problem 42E Chapter 1.4

Linear Algebra and Its Applications | 4th Edition

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Problem 42E

Problem 42E

[M] Find a column of the matrix in Exercise 40 that can be deleted and yet have the remaining matrix columns still span ℝ4. Can you delete more than one column?

Exercise 40:

Step-by-Step Solution:
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Mart 1 Lecture 16: The Cell Cycle ● Cell division functions in reproduction, growth, and repair. ○ The ability to reproduce distinguishes living things from nonliving. ○ The continuity of life is based on cell division. ■ Division of unicellular organisms reproduces an entire organism, while it produces progeny for multicellular organisms. ■ This is used to repair and renew cells that die. ■ Requires the distribution of identical genetic material, DNA, from generation to generation. ○ Genome ​ is a cell’s genetic information, packaged as DNA. ■ In prokaryotes, often a single long DNA molecules, while in eukaryotes, it consists of several DNA molecules. ○ DNA molecules are packaged into ​hromosomes.​ ■ Every eukaryotic organism has a characteristic number of chromosomes, humans have 46 chromosomes. ○ Each duplicated chromosomes: ■ Consists of twosister chromatid, contain identical copies of chromosome’s DNA. ● Mitosis ​is a type of cell division that results in two daughter cells each having the same number and kind of chromosomes as the parent nucleus. ○ Cytokinesis​ is the cytoplasmic division of a cell, bringing about the separation into two daughter cells. ○ Interphase has three subphases: ■ G₁ (“first gap”) is centered around growth, the increase in size. ■ S phase​ (“synthesis”) the chromosomes are copied. ■ G₂ (“second gap”) completes the preparations for cell division, the duplication of the centrosomes*. ■ M phase ​divides the cell, restarting the process. ○ Mitosis has five subphases: ■ Prophase​, the DNA becomes condensed into chromosomes and the centrosomes migrate to opposite sides of the cell. ■ Prometaphase​ , the nuclear envelope if fragmented and the microtubules begin to align to two spindle poles. ■ Metaphase​ , the chromosomes line up. ■ Anaphase​ , the sister chromatids are pulled apart. ■ Telophase​, the nuclear envelope is reformed and cytokinesis takes place (division of the cells). Mart 2 ○ Mitotic spindle is a structure composed of microtubules which segregates chromosomes into the daughter cells during mitosis. ■ The assembly of spindle microtubules starts in thcentrosome​. ■ Spindle microtubules attach tokinetochores of sister chromatids. ■ Spindle fibers shorten during anaphase from end attached to the chromosomes. ● Cytokinesis​ is the division of the cytoplasm, typically follows mitosis. ○ Contraction of actin­myosin ring pinches the cell in two. ○ In plant cells, during telophase, vesicles forcell plat. ○ A molecular control system drives the cell cycle through the use of specific chemical signals in the cytoplasm. ■ Checkpoints ​are critical control points where stop and go signals regulate the cell cycle, the most importantG₁s. ■ The ​G₀ are nondividing, they can later reenter the cell cycle. ○ Kinases ​are proteins that catalyse the transfer of a phosphate group from ATP to a specified molecule, present in a constant amount. ○ Cyclin​ are any of a number of proteins associated with the cycle of cell division that are thought to initiate certain processes of mitosis, proteins fluctuate cyclically. ○ MPF (mix of kinase and cyclin) triggers the cell’s passage past the checkpoint, this promotes mitosis by: ■ Phosphorylating other kinases. ■ Stimulating the breakdown of the nuclear envelope. ■ The breakdown of cyclin. ○ Growth factors ​are a substance, such as a vitamin or hormone, that is required for the stimulation of growth in living cells. ■ This triggers a signal­transduction pathway which leads to cell division. ■ Growth factors are a key indensity­dependent inhibitio of cell division ■ Most animal cells exhibit anchorage dependence ​for cell division. ● Cancer cells have escaped from cell cycle controls. ○ Transformation ​is the process of a normal cell converting to a cancer cell. ○ If not, a cell proliferates to fortumor, a mass of abnormal cells. ■ If the abnormal cells remain at the originating site ibenign tumor​. ■ A ​malignant tumor​ occurs when the abnormal cells leave the original site. ○ Metastasis ​is when cancer cells lose attachment to nearby cells, carried by blood and lymph system to other tissues, and start more tumors ■ Cellular transformation always involves the alteration of genes that influence the cell cycle control system. Mart 3

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[M] Find a column of the matrix in Exercise 40 that can be